Theorem addsdilem2 28111

Description: Lemma for surreal distribution. Expand the right hand side of the main expression. (Contributed by Scott Fenton, 8-Mar-2025.)

Hypotheses

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

Assertion

Ref

Expression

addsdilem2

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

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

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

Proof of Theorem addsdilem2

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

Step	Hyp	Ref	Expression
1		addsdilem.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ No )
2		addsdilem.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No )
3	1, 2	mulscut2 28092	. . 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		addsdilem.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ No )
5	1, 4	mulscut2 28092	. . 3 ⊢ (𝜑 → ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))}) <<s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))}))
6		mulsval2 28070	. . . 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 584	. . 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		mulsval2 28070	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s 𝐶) = (({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))}) |s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))})))
9	1, 4, 8	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴 ·s 𝐶) = (({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))}) |s ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))})))
10	3, 5, 7, 9	addsunif 27965	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) +s (𝐴 ·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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}) |s ({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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)})))
11		unab 4257	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))}
12		rexun 4145	. . . . . . 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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
13		eqeq1 2737	. . . . . . . . . . 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))))
14	13	2rexbidv 3198	. . . . . . . . . 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))))
15	14	rexab 3650	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
16		rexcom4 3260	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
17		rexcom4 3260	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
18		ovex 7388	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∈ V
19		oveq1 7362	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (𝑡 +s (𝐴 ·s 𝐶)) = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
20	19	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) → (𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ 𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))))
21	18, 20	ceqsexv 3487	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ 𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
22	21	rexbii 3080	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
23	17, 22	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
24	23	rexbii 3080	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
25		r19.41vv 3203	. . . . . . . . . . 11 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
26	25	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
27	16, 24, 26	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
28	15, 27	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
29		eqeq1 2737	. . . . . . . . . . 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))))
30	29	2rexbidv 3198	. . . . . . . . . 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))))
31	30	rexab 3650	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
32		rexcom4 3260	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
33		rexcom4 3260	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
34		ovex 7388	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∈ V
35		oveq1 7362	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (𝑡 +s (𝐴 ·s 𝐶)) = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
36	35	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) → (𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ 𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))))
37	34, 36	ceqsexv 3487	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ 𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
38	37	rexbii 3080	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
39	33, 38	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
40	39	rexbii 3080	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
41		r19.41vv 3203	. . . . . . . . . . 11 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
42	41	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
43	32, 40, 42	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
44	31, 43	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
45	28, 44	orbi12i 914	. . . . . . 7 ⊢ ((∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))))
46	12, 45	bitr2i 276	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))) ↔ ∃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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))
47	46	abbii 2800	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))}
48	11, 47	eqtri 2756	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))}
49		unab 4257	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))}
50		rexun 4145	. . . . . . 7 ⊢ (∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
51		eqeq1 2737	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ↔ 𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
52	51	2rexbidv 3198	. . . . . . . . . 10 ⊢ (𝑏 = 𝑡 → (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
53	52	rexab 3650	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
54		rexcom4 3260	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
55		rexcom4 3260	. . . . . . . . . . . 12 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
56		ovex 7388	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∈ V
57		oveq2 7363	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) → ((𝐴 ·s 𝐵) +s 𝑡) = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
58	57	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) → (𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))))
59	56, 58	ceqsexv 3487	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
60	59	rexbii 3080	. . . . . . . . . . . 12 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
61	55, 60	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
62	61	rexbii 3080	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
63		r19.41vv 3203	. . . . . . . . . . 11 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
64	63	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
65	54, 62, 64	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
66	53, 65	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))))
67		eqeq1 2737	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ↔ 𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
68	67	2rexbidv 3198	. . . . . . . . . 10 ⊢ (𝑏 = 𝑡 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
69	68	rexab 3650	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
70		rexcom4 3260	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
71		rexcom4 3260	. . . . . . . . . . . 12 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
72		ovex 7388	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∈ V
73		oveq2 7363	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) → ((𝐴 ·s 𝐵) +s 𝑡) = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
74	73	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) → (𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))))
75	72, 74	ceqsexv 3487	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
76	75	rexbii 3080	. . . . . . . . . . . 12 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
77	71, 76	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
78	77	rexbii 3080	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
79		r19.41vv 3203	. . . . . . . . . . 11 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
80	79	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
81	70, 78, 80	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
82	69, 81	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))
83	66, 82	orbi12i 914	. . . . . . 7 ⊢ ((∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))))
84	50, 83	bitr2i 276	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))
85	84	abbii 2800	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))))} = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}
86	49, 85	eqtri 2756	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))}) = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}
87	48, 86	uneq12i 4115	. . 3 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)})
88		unab 4257	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))}
89		rexun 4145	. . . . . . 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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
90		eqeq1 2737	. . . . . . . . . . 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))))
91	90	2rexbidv 3198	. . . . . . . . . 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))))
92	91	rexab 3650	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
93		rexcom4 3260	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
94		rexcom4 3260	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
95		ovex 7388	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∈ V
96		oveq1 7362	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (𝑡 +s (𝐴 ·s 𝐶)) = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
97	96	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) → (𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ 𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))))
98	95, 97	ceqsexv 3487	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ 𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
99	98	rexbii 3080	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
100	94, 99	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
101	100	rexbii 3080	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
102		r19.41vv 3203	. . . . . . . . . . 11 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
103	102	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
104	93, 101, 103	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
105	92, 104	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)))
106		eqeq1 2737	. . . . . . . . . . 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))))
107	106	2rexbidv 3198	. . . . . . . . . 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))))
108	107	rexab 3650	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
109		rexcom4 3260	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
110		rexcom4 3260	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
111		ovex 7388	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∈ V
112		oveq1 7362	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (𝑡 +s (𝐴 ·s 𝐶)) = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
113	112	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) → (𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ 𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))))
114	111, 113	ceqsexv 3487	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ 𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
115	114	rexbii 3080	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
116	110, 115	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
117	116	rexbii 3080	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
118		r19.41vv 3203	. . . . . . . . . . 11 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
119	118	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))))
120	109, 117, 119	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ∧ 𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
121	108, 120	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))
122	105, 121	orbi12i 914	. . . . . . 7 ⊢ ((∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿))}𝑎 = (𝑡 +s (𝐴 ·s 𝐶))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))))
123	89, 122	bitr2i 276	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))) ↔ ∃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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶)))
124	123	abbii 2800	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶)) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶)))} = {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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))}
125	88, 124	eqtri 2756	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))}) = {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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))}
126		unab 4257	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))}
127		rexun 4145	. . . . . . 7 ⊢ (∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
128		eqeq1 2737	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ↔ 𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
129	128	2rexbidv 3198	. . . . . . . . . 10 ⊢ (𝑏 = 𝑡 → (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
130	129	rexab 3650	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
131		rexcom4 3260	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
132		rexcom4 3260	. . . . . . . . . . . 12 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
133		ovex 7388	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∈ V
134		oveq2 7363	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) → ((𝐴 ·s 𝐵) +s 𝑡) = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
135	134	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) → (𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))))
136	133, 135	ceqsexv 3487	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
137	136	rexbii 3080	. . . . . . . . . . . 12 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
138	132, 137	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
139	138	rexbii 3080	. . . . . . . . . 10 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
140		r19.41vv 3203	. . . . . . . . . . 11 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
141	140	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
142	131, 139, 141	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
143	130, 142	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))))
144		eqeq1 2737	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑡 → (𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ↔ 𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
145	144	2rexbidv 3198	. . . . . . . . . 10 ⊢ (𝑏 = 𝑡 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
146	145	rexab 3650	. . . . . . . . 9 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
147		rexcom4 3260	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
148		rexcom4 3260	. . . . . . . . . . . 12 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
149		ovex 7388	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∈ V
150		oveq2 7363	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) → ((𝐴 ·s 𝐵) +s 𝑡) = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
151	150	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) → (𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))))
152	149, 151	ceqsexv 3487	. . . . . . . . . . . . 13 ⊢ (∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ 𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
153	152	rexbii 3080	. . . . . . . . . . . 12 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
154	148, 153	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
155	154	rexbii 3080	. . . . . . . . . 10 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
156		r19.41vv 3203	. . . . . . . . . . 11 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
157	156	exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑡∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)))
158	147, 155, 157	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑡(∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
159	146, 158	bitri 275	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))
160	143, 159	orbi12i 914	. . . . . . 7 ⊢ ((∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))}𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))))
161	127, 160	bitr2i 276	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡))
162	161	abbii 2800	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))))} = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}
163	126, 162	eqtri 2756	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))}) = {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}
164	125, 163	uneq12i 4115	. . 3 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)})
165	87, 164	oveq12i 7367	. 2 ⊢ ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}) |s ({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))})𝑎 = (𝑡 +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿))})𝑎 = ((𝐴 ·s 𝐵) +s 𝑡)}))
166	10, 165	eqtr4di 2786	1 ⊢ (𝜑 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))}))))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2711  ∃wrex 3057   ∪ cun 3896  ‘cfv 6489  (class class class)co 7355   No csur 27598   |s cscut 27742   L cleft 27806   R cright 27807   +_s cadds 27922   -_s csubs 27982   ·_s cmuls 28065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-1o 8394  df-2o 8395  df-nadd 8590  df-no 27601  df-slt 27602  df-bday 27603  df-sle 27704  df-sslt 27741  df-scut 27743  df-0s 27788  df-made 27808  df-old 27809  df-left 27811  df-right 27812  df-norec 27901  df-norec2 27912  df-adds 27923  df-negs 27983  df-subs 27984  df-muls 28066

This theorem is referenced by:  addsdi  28114