Theorem addsdilem1 28195

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		lltropt 27929	. . . 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	addscut2 28030	. . 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 27959	. . . . 5 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
9	8	eqcomd 2746	. . 3 ⊢ (𝜑 → 𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
10		addsval2 28014	. . . 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 583	. . 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 28194	. 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 4327	. . . . 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 3128	. . . . . . 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 4219	. . . . . . . . 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 2744	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝑦𝐿 +s 𝐶) ↔ 𝑏 = (𝑦𝐿 +s 𝐶)))
17	16	rexbidv 3185	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶)))
18	17	rexab 3716	. . . . . . . . . . 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 7481	. . . . . . . . . . . . . 14 ⊢ (𝑦𝐿 +s 𝐶) ∈ V
21		oveq2 7456	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝑦𝐿 +s 𝐶)))
22	21	oveq2d 7464	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))
24	22, 23	oveq12d 7466	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 3542	. . . . . . . . . . . . 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 3100	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . 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 1846	. . . . . . . . . . . 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 302	. . . . . . . . . . 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 275	. . . . . . . . . 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 2744	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝐵 +s 𝑧𝐿) ↔ 𝑏 = (𝐵 +s 𝑧𝐿)))
33	32	rexbidv 3185	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿)))
34	33	rexab 3716	. . . . . . . . . . 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 7481	. . . . . . . . . . . . . 14 ⊢ (𝐵 +s 𝑧𝐿) ∈ V
37		oveq2 7456	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝐵 +s 𝑧𝐿)))
38	37	oveq2d 7464	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))
40	38, 39	oveq12d 7466	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 3542	. . . . . . . . . . . . 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 3100	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . 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 1846	. . . . . . . . . . . 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 302	. . . . . . . . . . 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 275	. . . . . . . . . 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 913	. . . . . . . . 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 276	. . . . . . . 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 3100	. . . . . . 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 277	. . . . . 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 2812	. . . . 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 2768	. . . 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 4327	. . . . 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 3128	. . . . . . 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 4219	. . . . . . . . 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 2744	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝑦𝑅 +s 𝐶) ↔ 𝑏 = (𝑦𝑅 +s 𝐶)))
58	57	rexbidv 3185	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶)))
59	58	rexab 3716	. . . . . . . . . . 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 7481	. . . . . . . . . . . . . 14 ⊢ (𝑦𝑅 +s 𝐶) ∈ V
62		oveq2 7456	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝑦𝑅 +s 𝐶)))
63	62	oveq2d 7464	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))
65	63, 64	oveq12d 7466	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 3542	. . . . . . . . . . . . 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 3100	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . 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 1846	. . . . . . . . . . . 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 302	. . . . . . . . . . 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 275	. . . . . . . . . 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 2744	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝐵 +s 𝑧𝑅) ↔ 𝑏 = (𝐵 +s 𝑧𝑅)))
74	73	rexbidv 3185	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅)))
75	74	rexab 3716	. . . . . . . . . . 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 7481	. . . . . . . . . . . . . 14 ⊢ (𝐵 +s 𝑧𝑅) ∈ V
78		oveq2 7456	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝐵 +s 𝑧𝑅)))
79	78	oveq2d 7464	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))
81	79, 80	oveq12d 7466	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 3542	. . . . . . . . . . . . 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 3100	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . 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 1846	. . . . . . . . . . . 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 302	. . . . . . . . . . 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 275	. . . . . . . . . 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 913	. . . . . . . . 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 276	. . . . . . . 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 3100	. . . . . . 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 277	. . . . . 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 2812	. . . . 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 2768	. . . 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 4189	. . 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 4327	. . . . 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 3128	. . . . . . 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 4219	. . . . . . . . 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 3716	. . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))
103	101, 102	oveq12d 7466	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 3542	. . . . . . . . . . . . 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 3100	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . 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 1846	. . . . . . . . . . . 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 302	. . . . . . . . . . 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 275	. . . . . . . . . 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 3716	. . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))
115	113, 114	oveq12d 7466	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 3542	. . . . . . . . . . . . 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 3100	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . 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 1846	. . . . . . . . . . . 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 302	. . . . . . . . . . 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 275	. . . . . . . . . 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 913	. . . . . . . . 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 276	. . . . . . . 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 3100	. . . . . . 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 277	. . . . . 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 2812	. . . . 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 2768	. . . 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 4327	. . . . 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 3128	. . . . . . 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 4219	. . . . . . . . 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 3716	. . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))
136	134, 135	oveq12d 7466	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 3542	. . . . . . . . . . . . 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 3100	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . 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 1846	. . . . . . . . . . . 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 302	. . . . . . . . . . 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 275	. . . . . . . . . 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 3716	. . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))
148	146, 147	oveq12d 7466	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 3542	. . . . . . . . . . . . 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 3100	. . . . . . . . . . . 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 3195	. . . . . . . . . . . . 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 1846	. . . . . . . . . . . 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 302	. . . . . . . . . . 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 275	. . . . . . . . . 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 913	. . . . . . . . 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 276	. . . . . . . 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 3100	. . . . . . 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 277	. . . . . 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 2812	. . . . 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 2768	. . . 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 4189	. . 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 7460	. 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 2798	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)))}))))

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:  addsdi  28199