Theorem addsasslem1 28037

Description: Lemma for addition associativity. Expand one form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.)

Hypotheses

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

Assertion

Ref	Expression
addsasslem1	⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧   𝐵,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧   𝐶,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧   𝜑,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧

Proof of Theorem addsasslem1

Dummy variables 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addsasslem.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
2		addsasslem.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
3	1, 2	addscut 28012	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})))
4	3	simp2d 1143	. . . 4 ⊢ (𝜑 → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)})
5	3	simp3d 1144	. . . 4 ⊢ (𝜑 → {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
6		ovex 7465	. . . . . 6 ⊢ (𝐴 +s 𝐵) ∈ V
7	6	snnz 4775	. . . . 5 ⊢ {(𝐴 +s 𝐵)} ≠ ∅
8		sslttr 27853	. . . . 5 ⊢ ((({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}) ∧ {(𝐴 +s 𝐵)} ≠ ∅) → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
9	7, 8	mp3an3 1451	. . . 4 ⊢ ((({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})) → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
10	4, 5, 9	syl2anc 584	. . 3 ⊢ (𝜑 → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
11		lltropt 27912	. . . 4 ⊢ ( L ‘𝐶) <<s ( R ‘𝐶)
12	11	a1i 11	. . 3 ⊢ (𝜑 → ( L ‘𝐶) <<s ( R ‘𝐶))
13		addsval2 27997	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) |s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})))
14	1, 2, 13	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) |s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})))
15		addsasslem.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No )
16		lrcut 27942	. . . . 5 ⊢ (𝐶 ∈ No → (( L ‘𝐶) |s ( R ‘𝐶)) = 𝐶)
17	15, 16	syl 17	. . . 4 ⊢ (𝜑 → (( L ‘𝐶) |s ( R ‘𝐶)) = 𝐶)
18	17	eqcomd 2742	. . 3 ⊢ (𝜑 → 𝐶 = (( L ‘𝐶) |s ( R ‘𝐶)))
19	10, 12, 14, 18	addsunif 28036	. 2 ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = (({𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶)} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s ({𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})))
20		unab 4307	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑦 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)}) = {𝑦 ∣ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))}
21		eqeq1 2740	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑧 = ((𝐴 +s 𝑚) +s 𝐶) ↔ 𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
22	21	rexbidv 3178	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
23	22	cbvabv 2811	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)} = {𝑦 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)}
24	23	uneq2i 4164	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)}) = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑦 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)})
25		rexun 4195	. . . . . . 7 ⊢ (∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶) ↔ (∃ℎ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = (ℎ +s 𝐶) ∨ ∃ℎ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = (ℎ +s 𝐶)))
26		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑑 = ℎ → (𝑑 = (𝑙 +s 𝐵) ↔ ℎ = (𝑙 +s 𝐵)))
27	26	rexbidv 3178	. . . . . . . . . 10 ⊢ (𝑑 = ℎ → (∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵) ↔ ∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵)))
28	27	rexab 3699	. . . . . . . . 9 ⊢ (∃ℎ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = (ℎ +s 𝐶) ↔ ∃ℎ(∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)))
29		rexcom4 3287	. . . . . . . . . 10 ⊢ (∃𝑙 ∈ ( L ‘𝐴)∃ℎ(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃ℎ∃𝑙 ∈ ( L ‘𝐴)(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)))
30		ovex 7465	. . . . . . . . . . . 12 ⊢ (𝑙 +s 𝐵) ∈ V
31		oveq1 7439	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑙 +s 𝐵) → (ℎ +s 𝐶) = ((𝑙 +s 𝐵) +s 𝐶))
32	31	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑙 +s 𝐵) → (𝑦 = (ℎ +s 𝐶) ↔ 𝑦 = ((𝑙 +s 𝐵) +s 𝐶)))
33	30, 32	ceqsexv 3531	. . . . . . . . . . 11 ⊢ (∃ℎ(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ 𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
34	33	rexbii 3093	. . . . . . . . . 10 ⊢ (∃𝑙 ∈ ( L ‘𝐴)∃ℎ(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
35		r19.41v 3188	. . . . . . . . . . 11 ⊢ (∃𝑙 ∈ ( L ‘𝐴)(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ (∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)))
36	35	exbii 1847	. . . . . . . . . 10 ⊢ (∃ℎ∃𝑙 ∈ ( L ‘𝐴)(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃ℎ(∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)))
37	29, 34, 36	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃ℎ(∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
38	28, 37	bitri 275	. . . . . . . 8 ⊢ (∃ℎ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = (ℎ +s 𝐶) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
39		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑒 = ℎ → (𝑒 = (𝐴 +s 𝑚) ↔ ℎ = (𝐴 +s 𝑚)))
40	39	rexbidv 3178	. . . . . . . . . 10 ⊢ (𝑒 = ℎ → (∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚) ↔ ∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚)))
41	40	rexab 3699	. . . . . . . . 9 ⊢ (∃ℎ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = (ℎ +s 𝐶) ↔ ∃ℎ(∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)))
42		rexcom4 3287	. . . . . . . . . 10 ⊢ (∃𝑚 ∈ ( L ‘𝐵)∃ℎ(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃ℎ∃𝑚 ∈ ( L ‘𝐵)(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)))
43		ovex 7465	. . . . . . . . . . . 12 ⊢ (𝐴 +s 𝑚) ∈ V
44		oveq1 7439	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐴 +s 𝑚) → (ℎ +s 𝐶) = ((𝐴 +s 𝑚) +s 𝐶))
45	44	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐴 +s 𝑚) → (𝑦 = (ℎ +s 𝐶) ↔ 𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
46	43, 45	ceqsexv 3531	. . . . . . . . . . 11 ⊢ (∃ℎ(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ 𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
47	46	rexbii 3093	. . . . . . . . . 10 ⊢ (∃𝑚 ∈ ( L ‘𝐵)∃ℎ(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
48		r19.41v 3188	. . . . . . . . . . 11 ⊢ (∃𝑚 ∈ ( L ‘𝐵)(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ (∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)))
49	48	exbii 1847	. . . . . . . . . 10 ⊢ (∃ℎ∃𝑚 ∈ ( L ‘𝐵)(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃ℎ(∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)))
50	42, 47, 49	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃ℎ(∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
51	41, 50	bitri 275	. . . . . . . 8 ⊢ (∃ℎ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = (ℎ +s 𝐶) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
52	38, 51	orbi12i 914	. . . . . . 7 ⊢ ((∃ℎ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = (ℎ +s 𝐶) ∨ ∃ℎ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = (ℎ +s 𝐶)) ↔ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
53	25, 52	bitri 275	. . . . . 6 ⊢ (∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶) ↔ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
54	53	abbii 2808	. . . . 5 ⊢ {𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶)} = {𝑦 ∣ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))}
55	20, 24, 54	3eqtr4ri 2775	. . . 4 ⊢ {𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶)} = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)})
56	55	uneq1i 4163	. . 3 ⊢ ({𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶)} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)})
57		unab 4307	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑎 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)}) = {𝑎 ∣ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))}
58		eqeq1 2740	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 = ((𝐴 +s 𝑞) +s 𝐶) ↔ 𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
59	58	rexbidv 3178	. . . . . . 7 ⊢ (𝑏 = 𝑎 → (∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
60	59	cbvabv 2811	. . . . . 6 ⊢ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)} = {𝑎 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)}
61	60	uneq2i 4164	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)}) = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑎 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)})
62		rexun 4195	. . . . . . 7 ⊢ (∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶) ↔ (∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶)))
63		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑖 → (𝑓 = (𝑝 +s 𝐵) ↔ 𝑖 = (𝑝 +s 𝐵)))
64	63	rexbidv 3178	. . . . . . . . . 10 ⊢ (𝑓 = 𝑖 → (∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵)))
65	64	rexab 3699	. . . . . . . . 9 ⊢ (∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑖(∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
66		rexcom4 3287	. . . . . . . . . 10 ⊢ (∃𝑝 ∈ ( R ‘𝐴)∃𝑖(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖∃𝑝 ∈ ( R ‘𝐴)(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
67		ovex 7465	. . . . . . . . . . . 12 ⊢ (𝑝 +s 𝐵) ∈ V
68		oveq1 7439	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑝 +s 𝐵) → (𝑖 +s 𝐶) = ((𝑝 +s 𝐵) +s 𝐶))
69	68	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑝 +s 𝐵) → (𝑎 = (𝑖 +s 𝐶) ↔ 𝑎 = ((𝑝 +s 𝐵) +s 𝐶)))
70	67, 69	ceqsexv 3531	. . . . . . . . . . 11 ⊢ (∃𝑖(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ 𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
71	70	rexbii 3093	. . . . . . . . . 10 ⊢ (∃𝑝 ∈ ( R ‘𝐴)∃𝑖(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
72		r19.41v 3188	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ ( R ‘𝐴)(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ (∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
73	72	exbii 1847	. . . . . . . . . 10 ⊢ (∃𝑖∃𝑝 ∈ ( R ‘𝐴)(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖(∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
74	66, 71, 73	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑖(∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
75	65, 74	bitri 275	. . . . . . . 8 ⊢ (∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
76		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑖 → (𝑔 = (𝐴 +s 𝑞) ↔ 𝑖 = (𝐴 +s 𝑞)))
77	76	rexbidv 3178	. . . . . . . . . 10 ⊢ (𝑔 = 𝑖 → (∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞)))
78	77	rexab 3699	. . . . . . . . 9 ⊢ (∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
79		rexcom4 3287	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
80		ovex 7465	. . . . . . . . . . . 12 ⊢ (𝐴 +s 𝑞) ∈ V
81		oveq1 7439	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐴 +s 𝑞) → (𝑖 +s 𝐶) = ((𝐴 +s 𝑞) +s 𝐶))
82	81	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐴 +s 𝑞) → (𝑎 = (𝑖 +s 𝐶) ↔ 𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
83	80, 82	ceqsexv 3531	. . . . . . . . . . 11 ⊢ (∃𝑖(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ 𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
84	83	rexbii 3093	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
85		r19.41v 3188	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ (∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
86	85	exbii 1847	. . . . . . . . . 10 ⊢ (∃𝑖∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
87	79, 84, 86	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
88	78, 87	bitri 275	. . . . . . . 8 ⊢ (∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
89	75, 88	orbi12i 914	. . . . . . 7 ⊢ ((∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶)) ↔ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
90	62, 89	bitri 275	. . . . . 6 ⊢ (∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶) ↔ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
91	90	abbii 2808	. . . . 5 ⊢ {𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} = {𝑎 ∣ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))}
92	57, 61, 91	3eqtr4ri 2775	. . . 4 ⊢ {𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)})
93	92	uneq1i 4163	. . 3 ⊢ ({𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)}) = (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})
94	56, 93	oveq12i 7444	. 2 ⊢ (({𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶)} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s ({𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)}))
95	19, 94	eqtrdi 2792	1 ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2713   ≠ wne 2939  ∃wrex 3069   ∪ cun 3948  ∅c0 4332  {csn 4625   class class class wbr 5142  ‘cfv 6560  (class class class)co 7432   No csur 27685   <<s csslt 27826   |s cscut 27828   L cleft 27885   R cright 27886   +_s cadds 27993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec2 27983  df-adds 27994

This theorem is referenced by:  addsass  28039