Theorem addsdilem1 28373

Description: Lemma for surreal distribution. Expand the left 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

addsdilem1

⊢ (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +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 addsdilem1

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

Step	Hyp	Ref	Expression
1		lltr 28084	. . . 4 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
3		addsdilem.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No )
4		addsdilem.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ No )
5	3, 4	addcuts2 28201	. . 3 ⊢ (𝜑 → ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}) <<s ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}))
6		addsdilem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ No )
7		lrcut 28126	. . . . 5 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
8	6, 7	syl 18	. . . 4 ⊢ (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
9	8	eqcomd 2771	. . 3 ⊢ (𝜑 → 𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
10		addsval2 28185	. . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 +s 𝐶) = (({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}) |s ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})))
11	3, 4, 10	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐵 +s 𝐶) = (({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}) |s ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})))
12	2, 5, 9, 11	mulsunif 28372	. 2 ⊢ (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}) |s ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))})))
13		unab 4261	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))}
14		r19.43 3135	. . . . . . 7 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))))
15		rexun 4149	. . . . . . . . 9 ⊢ (∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
16		eqeq1 2769	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝑦𝐿 +s 𝐶) ↔ 𝑏 = (𝑦𝐿 +s 𝐶)))
17	16	rexbidv 3191	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶)))
18	17	rexab 3660	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
19		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
20		ovex 7449	. . . . . . . . . . . . . 14 ⊢ (𝑦𝐿 +s 𝐶) ∈ V
21		oveq2 7424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝑦𝐿 +s 𝐶)))
22	21	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))))
23		oveq2 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))
24	22, 23	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
25	24	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))))
26	20, 25	ceqsexv 3505	. . . . . . . . . . . . 13 ⊢ (∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
27	26	rexbii 3114	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
28		r19.41v 3197	. . . . . . . . . . . . 13 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
29	28	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
30	19, 27, 29	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
31	18, 30	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
32		eqeq1 2769	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝐵 +s 𝑧𝐿) ↔ 𝑏 = (𝐵 +s 𝑧𝐿)))
33	32	rexbidv 3191	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿)))
34	33	rexab 3660	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
35		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
36		ovex 7449	. . . . . . . . . . . . . 14 ⊢ (𝐵 +s 𝑧𝐿) ∈ V
37		oveq2 7424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝐵 +s 𝑧𝐿)))
38	37	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))))
39		oveq2 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))
40	38, 39	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
41	40	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))))
42	36, 41	ceqsexv 3505	. . . . . . . . . . . . 13 ⊢ (∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
43	42	rexbii 3114	. . . . . . . . . . . 12 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
44		r19.41v 3197	. . . . . . . . . . . . 13 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
45	44	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
46	35, 43, 45	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
47	34, 46	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
48	31, 47	orbi12i 928	. . . . . . . . 9 ⊢ ((∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))))
49	15, 48	bitr2i 279	. . . . . . . 8 ⊢ ((∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
50	49	rexbii 3114	. . . . . . 7 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
51	14, 50	bitr3i 280	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
52	51	abbii 2832	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))}
53	13, 52	eqtri 2788	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))}
54		unab 4261	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))}) = {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))}
55		r19.43 3135	. . . . . . 7 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))))
56		rexun 4149	. . . . . . . . 9 ⊢ (∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
57		eqeq1 2769	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝑦𝑅 +s 𝐶) ↔ 𝑏 = (𝑦𝑅 +s 𝐶)))
58	57	rexbidv 3191	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶)))
59	58	rexab 3660	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
60		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
61		ovex 7449	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑅 +s 𝐶) ∈ V
62		oveq2 7424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝑦𝑅 +s 𝐶)))
63	62	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))))
64		oveq2 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))
65	63, 64	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
66	65	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))))
67	61, 66	ceqsexv 3505	. . . . . . . . . . . . 13 ⊢ (∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
68	67	rexbii 3114	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
69		r19.41v 3197	. . . . . . . . . . . . 13 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
70	69	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
71	60, 68, 70	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
72	59, 71	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
73		eqeq1 2769	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝐵 +s 𝑧𝑅) ↔ 𝑏 = (𝐵 +s 𝑧𝑅)))
74	73	rexbidv 3191	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅)))
75	74	rexab 3660	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
76		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
77		ovex 7449	. . . . . . . . . . . . . 14 ⊢ (𝐵 +s 𝑧𝑅) ∈ V
78		oveq2 7424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝐵 +s 𝑧𝑅)))
79	78	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))))
80		oveq2 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))
81	79, 80	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
82	81	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))))
83	77, 82	ceqsexv 3505	. . . . . . . . . . . . 13 ⊢ (∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
84	83	rexbii 3114	. . . . . . . . . . . 12 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
85		r19.41v 3197	. . . . . . . . . . . . 13 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
86	85	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
87	76, 84, 86	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
88	75, 87	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
89	72, 88	orbi12i 928	. . . . . . . . 9 ⊢ ((∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))))
90	56, 89	bitr2i 279	. . . . . . . 8 ⊢ ((∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
91	90	rexbii 3114	. . . . . . 7 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
92	55, 91	bitr3i 280	. . . . . 6 ⊢ ((∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
93	92	abbii 2832	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}
94	54, 93	eqtri 2788	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))}) = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}
95	53, 94	uneq12i 4120	. . 3 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))})
96		unab 4261	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))}
97		r19.43 3135	. . . . . . 7 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))))
98		rexun 4149	. . . . . . . . 9 ⊢ (∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
99	58	rexab 3660	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
100		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
101	62	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))))
102		oveq2 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))
103	101, 102	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
104	103	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))))
105	61, 104	ceqsexv 3505	. . . . . . . . . . . . 13 ⊢ (∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
106	105	rexbii 3114	. . . . . . . . . . . 12 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
107		r19.41v 3197	. . . . . . . . . . . . 13 ⊢ (∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
108	107	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
109	100, 106, 108	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
110	99, 109	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
111	74	rexab 3660	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
112		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
113	78	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))))
114		oveq2 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))
115	113, 114	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
116	115	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))))
117	77, 116	ceqsexv 3505	. . . . . . . . . . . . 13 ⊢ (∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
118	117	rexbii 3114	. . . . . . . . . . . 12 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
119		r19.41v 3197	. . . . . . . . . . . . 13 ⊢ (∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
120	119	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
121	112, 118, 120	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
122	111, 121	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
123	110, 122	orbi12i 928	. . . . . . . . 9 ⊢ ((∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))))
124	98, 123	bitr2i 279	. . . . . . . 8 ⊢ ((∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
125	124	rexbii 3114	. . . . . . 7 ⊢ (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
126	97, 125	bitr3i 280	. . . . . 6 ⊢ ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
127	126	abbii 2832	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))}
128	96, 127	eqtri 2788	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))}
129		unab 4261	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}) = {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))}
130		r19.43 3135	. . . . . . 7 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))))
131		rexun 4149	. . . . . . . . 9 ⊢ (∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
132	17	rexab 3660	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
133		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
134	21	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))))
135		oveq2 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))
136	134, 135	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
137	136	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))))
138	20, 137	ceqsexv 3505	. . . . . . . . . . . . 13 ⊢ (∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
139	138	rexbii 3114	. . . . . . . . . . . 12 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
140		r19.41v 3197	. . . . . . . . . . . . 13 ⊢ (∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
141	140	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
142	133, 139, 141	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
143	132, 142	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
144	33	rexab 3660	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
145		rexcom4 3294	. . . . . . . . . . . 12 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
146	37	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))))
147		oveq2 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))
148	146, 147	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
149	148	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))))
150	36, 149	ceqsexv 3505	. . . . . . . . . . . . 13 ⊢ (∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
151	150	rexbii 3114	. . . . . . . . . . . 12 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
152		r19.41v 3197	. . . . . . . . . . . . 13 ⊢ (∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
153	152	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑏∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
154	145, 151, 153	3bitr3ri 305	. . . . . . . . . . 11 ⊢ (∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
155	144, 154	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
156	143, 155	orbi12i 928	. . . . . . . . 9 ⊢ ((∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))))
157	131, 156	bitr2i 279	. . . . . . . 8 ⊢ ((∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
158	157	rexbii 3114	. . . . . . 7 ⊢ (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
159	130, 158	bitr3i 280	. . . . . 6 ⊢ ((∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
160	159	abbii 2832	. . . . 5 ⊢ {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}
161	129, 160	eqtri 2788	. . . 4 ⊢ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}) = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}
162	128, 161	uneq12i 4120	. . 3 ⊢ (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))})) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))})
163	95, 162	oveq12i 7428	. 2 ⊢ ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}))) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}) |s ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}))
164	12, 163	eqtr4di 2818	1 ⊢ (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  ∃wrex 3091   ∪ cun 3904   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416   No csur 27833   <<s cslts 27979   |s ccuts 27981   L cleft 28047   R cright 28048   +_s cadds 28181   -_s csubs 28242   ·_s cmuls 28328

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-1o 8455  df-2o 8456  df-nadd 8654  df-no 27836  df-lts 27837  df-bday 27838  df-les 27938  df-slts 27980  df-cuts 27982  df-0s 28029  df-made 28049  df-old 28050  df-left 28052  df-right 28053  df-norec 28160  df-norec2 28171  df-adds 28182  df-negs 28243  df-subs 28244  df-muls 28329

This theorem is used by:  addsdi  28377