Theorem mulsunif2 28317

Description: Alternate expression for surreal multiplication. Note from [Conway] p. 19. (Contributed by Scott Fenton, 16-Mar-2025.)

Hypotheses

Ref	Expression
mulsunif2.1	⊢ (𝜑 → 𝐿 <<s 𝑅)
mulsunif2.2	⊢ (𝜑 → 𝑀 <<s 𝑆)
mulsunif2.3	⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
mulsunif2.4	⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))

Assertion

Ref	Expression
mulsunif2	⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))})))

Distinct variable groups:   𝐴,𝑎,𝑝,𝑞   𝐴,𝑏,𝑟,𝑠   𝐴,𝑐,𝑡,𝑢   𝐴,𝑑,𝑣,𝑤   𝐵,𝑎,𝑝,𝑞   𝐵,𝑏,𝑟,𝑠   𝐵,𝑐,𝑡,𝑢   𝐵,𝑑,𝑣,𝑤   𝐿,𝑎,𝑝   𝐿,𝑐,𝑡   𝑀,𝑎,𝑝,𝑞   𝑀,𝑑,𝑣,𝑤   𝑅,𝑏   𝑅,𝑑   𝑅,𝑟   𝑣,𝑅   𝑆,𝑏   𝑆,𝑐   𝑆,𝑟,𝑠   𝑡,𝑆,𝑢

Allowed substitution hints:   𝜑(𝑤,𝑣,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)   𝑅(𝑤,𝑢,𝑡,𝑠,𝑞,𝑝,𝑎,𝑐)   𝑆(𝑤,𝑣,𝑞,𝑝,𝑎,𝑑)   𝐿(𝑤,𝑣,𝑢,𝑠,𝑟,𝑞,𝑏,𝑑)   𝑀(𝑢,𝑡,𝑠,𝑟,𝑏,𝑐)

Proof of Theorem mulsunif2

Dummy variables 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 𝑘 𝑙 𝑚 𝑛 𝑜 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulsunif2.1	. . 3 ⊢ (𝜑 → 𝐿 <<s 𝑅)
2		mulsunif2.2	. . 3 ⊢ (𝜑 → 𝑀 <<s 𝑆)
3		mulsunif2.3	. . 3 ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
4		mulsunif2.4	. . 3 ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))
5	1, 2, 3, 4	mulsunif2lem 28316	. 2 ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} ∪ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))}) |s ({𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} ∪ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))})))
6		eqeq1 2769	. . . . . . 7 ⊢ (𝑒 = 𝑎 → (𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))))
7	6	2rexbidv 3230	. . . . . 6 ⊢ (𝑒 = 𝑎 → (∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))))
8		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑖 = 𝑝 → (𝐴 -s 𝑖) = (𝐴 -s 𝑝))
9	8	oveq1d 7415	. . . . . . . . 9 ⊢ (𝑖 = 𝑝 → ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)) = ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)))
10	9	oveq2d 7416	. . . . . . . 8 ⊢ (𝑖 = 𝑝 → ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))))
11	10	eqeq2d 2776	. . . . . . 7 ⊢ (𝑖 = 𝑝 → (𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)))))
12		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑗 = 𝑞 → (𝐵 -s 𝑗) = (𝐵 -s 𝑞))
13	12	oveq2d 7416	. . . . . . . . 9 ⊢ (𝑗 = 𝑞 → ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)) = ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))
14	13	oveq2d 7416	. . . . . . . 8 ⊢ (𝑗 = 𝑞 → ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))) = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞))))
15	14	eqeq2d 2776	. . . . . . 7 ⊢ (𝑗 = 𝑞 → (𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))) ↔ 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))))
16	11, 15	cbvrex2vw 3248	. . . . . 6 ⊢ (∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞))))
17	7, 16	bitrdi 290	. . . . 5 ⊢ (𝑒 = 𝑎 → (∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))))
18	17	cbvabv 2835	. . . 4 ⊢ {𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} = {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))}
19		eqeq1 2769	. . . . . . 7 ⊢ (𝑓 = 𝑏 → (𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))))
20	19	2rexbidv 3230	. . . . . 6 ⊢ (𝑓 = 𝑏 → (∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))))
21		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑘 = 𝑟 → (𝑘 -s 𝐴) = (𝑟 -s 𝐴))
22	21	oveq1d 7415	. . . . . . . . 9 ⊢ (𝑘 = 𝑟 → ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)) = ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)))
23	22	oveq2d 7416	. . . . . . . 8 ⊢ (𝑘 = 𝑟 → ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))))
24	23	eqeq2d 2776	. . . . . . 7 ⊢ (𝑘 = 𝑟 → (𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)))))
25		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑙 = 𝑠 → (𝑙 -s 𝐵) = (𝑠 -s 𝐵))
26	25	oveq2d 7416	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)) = ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))
27	26	oveq2d 7416	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))) = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵))))
28	27	eqeq2d 2776	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))))
29	24, 28	cbvrex2vw 3248	. . . . . 6 ⊢ (∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵))))
30	20, 29	bitrdi 290	. . . . 5 ⊢ (𝑓 = 𝑏 → (∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))))
31	30	cbvabv 2835	. . . 4 ⊢ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))} = {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}
32	18, 31	uneq12i 4122	. . 3 ⊢ ({𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} ∪ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))}) = ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))})
33		eqeq1 2769	. . . . . . 7 ⊢ (𝑔 = 𝑐 → (𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))))
34	33	2rexbidv 3230	. . . . . 6 ⊢ (𝑔 = 𝑐 → (∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))))
35		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑚 = 𝑡 → (𝐴 -s 𝑚) = (𝐴 -s 𝑡))
36	35	oveq1d 7415	. . . . . . . . 9 ⊢ (𝑚 = 𝑡 → ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)) = ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)))
37	36	oveq2d 7416	. . . . . . . 8 ⊢ (𝑚 = 𝑡 → ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))))
38	37	eqeq2d 2776	. . . . . . 7 ⊢ (𝑚 = 𝑡 → (𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)))))
39		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑛 = 𝑢 → (𝑛 -s 𝐵) = (𝑢 -s 𝐵))
40	39	oveq2d 7416	. . . . . . . . 9 ⊢ (𝑛 = 𝑢 → ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)) = ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))
41	40	oveq2d 7416	. . . . . . . 8 ⊢ (𝑛 = 𝑢 → ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))) = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))))
42	41	eqeq2d 2776	. . . . . . 7 ⊢ (𝑛 = 𝑢 → (𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))))
43	38, 42	cbvrex2vw 3248	. . . . . 6 ⊢ (∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))))
44	34, 43	bitrdi 290	. . . . 5 ⊢ (𝑔 = 𝑐 → (∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))))
45	44	cbvabv 2835	. . . 4 ⊢ {𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} = {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))}
46		eqeq1 2769	. . . . . . 7 ⊢ (ℎ = 𝑑 → (ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))))
47	46	2rexbidv 3230	. . . . . 6 ⊢ (ℎ = 𝑑 → (∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))))
48		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑜 = 𝑣 → (𝑜 -s 𝐴) = (𝑣 -s 𝐴))
49	48	oveq1d 7415	. . . . . . . . 9 ⊢ (𝑜 = 𝑣 → ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)) = ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)))
50	49	oveq2d 7416	. . . . . . . 8 ⊢ (𝑜 = 𝑣 → ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))))
51	50	eqeq2d 2776	. . . . . . 7 ⊢ (𝑜 = 𝑣 → (𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)))))
52		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝐵 -s 𝑥) = (𝐵 -s 𝑤))
53	52	oveq2d 7416	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)) = ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))
54	53	oveq2d 7416	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))) = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤))))
55	54	eqeq2d 2776	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))))
56	51, 55	cbvrex2vw 3248	. . . . . 6 ⊢ (∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤))))
57	47, 56	bitrdi 290	. . . . 5 ⊢ (ℎ = 𝑑 → (∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))))
58	57	cbvabv 2835	. . . 4 ⊢ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))} = {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))}
59	45, 58	uneq12i 4122	. . 3 ⊢ ({𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} ∪ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))}) = ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))})
60	32, 59	oveq12i 7412	. 2 ⊢ (({𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} ∪ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))}) |s ({𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} ∪ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))})) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))}))
61	5, 60	eqtrdi 2816	1 ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))})))

Syntax hints:   → wi 4   = wceq 1563  {cab 2743  ∃wrex 3089   ∪ cun 3905   class class class wbr 5104  (class class class)co 7400   <<s cslts 27904   |s ccuts 27906   +_s cadds 28106   -_s csubs 28167   ·_s cmuls 28253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27761  df-lts 27762  df-bday 27763  df-les 27863  df-slts 27905  df-cuts 27907  df-0s 27954  df-made 27974  df-old 27975  df-left 27977  df-right 27978  df-norec 28085  df-norec2 28096  df-adds 28107  df-negs 28168  df-subs 28169  df-muls 28254

This theorem is referenced by:  remulscl  28649