Theorem mulsasslem1 28207

Description: Lemma for mulsass 28210. Expand the left hand side of the formula. (Contributed by Scott Fenton, 9-Mar-2025.)

Hypotheses

Ref	Expression
mulsasslem.1	⊢ (𝜑 → 𝐴 ∈ No )
mulsasslem.2	⊢ (𝜑 → 𝐵 ∈ No )
mulsasslem.3	⊢ (𝜑 → 𝐶 ∈ No )

Assertion

Ref

Expression

mulsasslem1

⊢ (𝜑 → ((𝐴 ·s 𝐵) ·s 𝐶) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}))))

Distinct variable groups:   𝐴,𝑎,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅,𝑧_𝐿,𝑧_𝑅   𝐵,𝑎,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅,𝑧_𝐿,𝑧_𝑅   𝐶,𝑎,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅,𝑧_𝐿,𝑧_𝑅

Allowed substitution hints:   𝜑(𝑎,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅,𝑧_𝐿,𝑧_𝑅)

Proof of Theorem mulsasslem1

Dummy variables 𝑏 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulsasslem.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ No )
2		mulsasslem.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No )
3	1, 2	mulscut2 28177	. . 3 ⊢ (𝜑 → ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}) <<s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}))
4		lltropt 27929	. . . 4 ⊢ ( L ‘𝐶) <<s ( R ‘𝐶)
5	4	a1i 11	. . 3 ⊢ (𝜑 → ( L ‘𝐶) <<s ( R ‘𝐶))
6		mulsval2 28155	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) = (({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}) |s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})))
7	1, 2, 6	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}) |s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})))
8		mulsasslem.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No )
9		lrcut 27959	. . . . 5 ⊢ (𝐶 ∈ No → (( L ‘𝐶) |s ( R ‘𝐶)) = 𝐶)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → (( L ‘𝐶) |s ( R ‘𝐶)) = 𝐶)
11	10	eqcomd 2746	. . 3 ⊢ (𝜑 → 𝐶 = (( L ‘𝐶) |s ( R ‘𝐶)))
12	3, 5, 7, 11	mulsunif 28194	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) ·s 𝐶) = (({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))}) |s ({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))})))
13		unab 4327	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)))}
14		rexun 4219	. . . . . . 7 ⊢ (∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
15		eqeq1 2744	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ↔ 𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))))
16	15	2rexbidv 3228	. . . . . . . . . 10 ⊢ (𝑏 = 𝑡 → (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))))
17	16	rexab 3716	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
18		rexcom4 3294	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
19		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
20		ovex 7481	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∈ V
21		oveq1 7455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (𝑡 ·s 𝐶) = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶))
22	21	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → ((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) = (((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)))
23		oveq1 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (𝑡 ·s 𝑧𝐿) = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))
24	22, 23	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
25	24	eqeq2d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ 𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))))
26	25	rexbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))))
27	20, 26	ceqsexv 3542	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
28	27	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
29	19, 28	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
30	29	rexbii 3100	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
31		r19.41vv 3233	. . . . . . . . . . 11 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
32	31	exbii 1846	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
33	18, 30, 32	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
34	17, 33	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
35		eqeq1 2744	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ↔ 𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))))
36	35	2rexbidv 3228	. . . . . . . . . 10 ⊢ (𝑏 = 𝑡 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))))
37	36	rexab 3716	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
38		rexcom4 3294	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
39		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
40		ovex 7481	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∈ V
41		oveq1 7455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (𝑡 ·s 𝐶) = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶))
42	41	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → ((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) = (((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)))
43		oveq1 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (𝑡 ·s 𝑧𝐿) = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))
44	42, 43	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
45	44	eqeq2d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ 𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))))
46	45	rexbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))))
47	40, 46	ceqsexv 3542	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
48	47	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
49	39, 48	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
50	49	rexbii 3100	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
51		r19.41vv 3233	. . . . . . . . . . 11 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
52	51	exbii 1846	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
53	38, 50, 52	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
54	37, 53	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
55	34, 54	orbi12i 913	. . . . . . 7 ⊢ ((∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))))
56	14, 55	bitr2i 276	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)))
57	56	abbii 2812	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿)))} = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))}
58	13, 57	eqtri 2768	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))}
59		unab 4327	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)))}
60		rexun 4219	. . . . . . 7 ⊢ (∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
61		eqeq1 2744	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ↔ 𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))))
62	61	2rexbidv 3228	. . . . . . . . . 10 ⊢ (𝑏 = 𝑡 → (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))))
63	62	rexab 3716	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
64		rexcom4 3294	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
65		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
66		ovex 7481	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∈ V
67		oveq1 7455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (𝑡 ·s 𝐶) = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶))
68	67	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → ((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) = (((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)))
69		oveq1 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (𝑡 ·s 𝑧𝑅) = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))
70	68, 69	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
71	70	eqeq2d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ 𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))))
72	71	rexbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))))
73	66, 72	ceqsexv 3542	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
74	73	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
75	65, 74	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
76	75	rexbii 3100	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
77		r19.41vv 3233	. . . . . . . . . . 11 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
78	77	exbii 1846	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
79	64, 76, 78	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
80	63, 79	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
81		eqeq1 2744	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ↔ 𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))))
82	81	2rexbidv 3228	. . . . . . . . . 10 ⊢ (𝑏 = 𝑡 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))))
83	82	rexab 3716	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
84		rexcom4 3294	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
85		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
86		ovex 7481	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∈ V
87		oveq1 7455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (𝑡 ·s 𝐶) = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶))
88	87	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → ((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) = (((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)))
89		oveq1 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (𝑡 ·s 𝑧𝑅) = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))
90	88, 89	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
91	90	eqeq2d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ 𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))))
92	91	rexbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))))
93	86, 92	ceqsexv 3542	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
94	93	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
95	85, 94	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
96	95	rexbii 3100	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
97		r19.41vv 3233	. . . . . . . . . . 11 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
98	97	exbii 1846	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
99	84, 96, 98	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
100	83, 99	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
101	80, 100	orbi12i 913	. . . . . . 7 ⊢ ((∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))))
102	60, 101	bitr2i 276	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)))
103	102	abbii 2812	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅)))} = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))}
104	59, 103	eqtri 2768	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))}) = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))}
105	58, 104	uneq12i 4189	. . 3 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) = ({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))})
106		unab 4327	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)))}
107		rexun 4219	. . . . . . 7 ⊢ (∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
108	16	rexab 3716	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
109		rexcom4 3294	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
110		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
111	21	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → ((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) = (((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)))
112		oveq1 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (𝑡 ·s 𝑧𝑅) = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))
113	111, 112	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
114	113	eqeq2d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ 𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))))
115	114	rexbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))))
116	20, 115	ceqsexv 3542	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
117	116	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
118	110, 117	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
119	118	rexbii 3100	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
120		r19.41vv 3233	. . . . . . . . . . 11 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
121	120	exbii 1846	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
122	109, 119, 121	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
123	108, 122	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)))
124	36	rexab 3716	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
125		rexcom4 3294	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
126		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
127	41	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → ((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) = (((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)))
128		oveq1 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (𝑡 ·s 𝑧𝑅) = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))
129	127, 128	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
130	129	eqeq2d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ 𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))))
131	130	rexbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))))
132	40, 131	ceqsexv 3542	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
133	132	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
134	126, 133	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
135	134	rexbii 3100	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
136		r19.41vv 3233	. . . . . . . . . . 11 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
137	136	exbii 1846	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))))
138	125, 135, 137	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
139	124, 138	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)))
140	123, 139	orbi12i 913	. . . . . . 7 ⊢ ((∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))))
141	107, 140	bitr2i 276	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅)))
142	141	abbii 2812	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅)))} = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))}
143	106, 142	eqtri 2768	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))}
144		unab 4327	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)))}
145		rexun 4219	. . . . . . 7 ⊢ (∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
146	62	rexab 3716	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
147		rexcom4 3294	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
148		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
149	67	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → ((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) = (((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)))
150		oveq1 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (𝑡 ·s 𝑧𝐿) = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))
151	149, 150	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
152	151	eqeq2d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ 𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))))
153	152	rexbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))))
154	66, 153	ceqsexv 3542	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
155	154	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
156	148, 155	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
157	156	rexbii 3100	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
158		r19.41vv 3233	. . . . . . . . . . 11 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
159	158	exbii 1846	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
160	147, 157, 159	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
161	146, 160	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)))
162	82	rexab 3716	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
163		rexcom4 3294	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
164		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
165	87	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → ((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) = (((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)))
166		oveq1 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (𝑡 ·s 𝑧𝐿) = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))
167	165, 166	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
168	167	eqeq2d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ 𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))))
169	168	rexbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))))
170	86, 169	ceqsexv 3542	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
171	170	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
172	164, 171	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
173	172	rexbii 3100	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
174		r19.41vv 3233	. . . . . . . . . . 11 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
175	174	exbii 1846	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))))
176	163, 173, 175	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
177	162, 176	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)))
178	161, 177	orbi12i 913	. . . . . . 7 ⊢ ((∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))))
179	145, 178	bitr2i 276	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿)))
180	179	abbii 2812	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿)))} = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))}
181	144, 180	eqtri 2768	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}) = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))}
182	143, 181	uneq12i 4189	. . 3 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))})) = ({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))})
183	105, 182	oveq12i 7460	. 2 ⊢ ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}))) = (({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))}) |s ({𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))})∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s (𝑡 ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))})∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑡 ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s (𝑡 ·s 𝑧𝐿))}))
184	12, 183	eqtr4di 2798	1 ⊢ (𝜑 → ((𝐴 ·s 𝐵) ·s 𝐶) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}))))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∃wrex 3076   ∪ cun 3974   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448   No csur 27702   <<s csslt 27843   |s cscut 27845   L cleft 27902   R cright 27903   +_s cadds 28010   -_s csubs 28070   ·_s cmuls 28150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151

This theorem is referenced by:  mulsass  28210