Theorem mulsunif2 28563

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 28562	. 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 2764	. . . . . . 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 3227	. . . . . 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 7430	. . . . . . . . . 10 ⊢ (𝑖 = 𝑝 → (𝐴 −s 𝑖) = (𝐴 −s 𝑝))
9	8	oveq1d 7437	. . . . . . . . 9 ⊢ (𝑖 = 𝑝 → ((𝐴 −s 𝑖) ·s (𝐵 −s 𝑗)) = ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑗)))
10	9	oveq2d 7438	. . . . . . . 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 2771	. . . . . . 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 7430	. . . . . . . . . 10 ⊢ (𝑗 = 𝑞 → (𝐵 −s 𝑗) = (𝐵 −s 𝑞))
13	12	oveq2d 7438	. . . . . . . . 9 ⊢ (𝑗 = 𝑞 → ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑗)) = ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)))
14	13	oveq2d 7438	. . . . . . . 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 2771	. . . . . . 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 3245	. . . . . 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 2830	. . . 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 2764	. . . . . . 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 3227	. . . . . 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 7429	. . . . . . . . . 10 ⊢ (𝑘 = 𝑟 → (𝑘 −s 𝐴) = (𝑟 −s 𝐴))
22	21	oveq1d 7437	. . . . . . . . 9 ⊢ (𝑘 = 𝑟 → ((𝑘 −s 𝐴) ·s (𝑙 −s 𝐵)) = ((𝑟 −s 𝐴) ·s (𝑙 −s 𝐵)))
23	22	oveq2d 7438	. . . . . . . 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 2771	. . . . . . 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 7429	. . . . . . . . . 10 ⊢ (𝑙 = 𝑠 → (𝑙 −s 𝐵) = (𝑠 −s 𝐵))
26	25	oveq2d 7438	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → ((𝑟 −s 𝐴) ·s (𝑙 −s 𝐵)) = ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)))
27	26	oveq2d 7438	. . . . . . . 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 2771	. . . . . . 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 3245	. . . . . 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 2830	. . . 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 4164	. . 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 2764	. . . . . . 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 3227	. . . . . 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 7430	. . . . . . . . . 10 ⊢ (𝑚 = 𝑡 → (𝐴 −s 𝑚) = (𝐴 −s 𝑡))
36	35	oveq1d 7437	. . . . . . . . 9 ⊢ (𝑚 = 𝑡 → ((𝐴 −s 𝑚) ·s (𝑛 −s 𝐵)) = ((𝐴 −s 𝑡) ·s (𝑛 −s 𝐵)))
37	36	oveq2d 7438	. . . . . . . 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 2771	. . . . . . 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 7429	. . . . . . . . . 10 ⊢ (𝑛 = 𝑢 → (𝑛 −s 𝐵) = (𝑢 −s 𝐵))
40	39	oveq2d 7438	. . . . . . . . 9 ⊢ (𝑛 = 𝑢 → ((𝐴 −s 𝑡) ·s (𝑛 −s 𝐵)) = ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)))
41	40	oveq2d 7438	. . . . . . . 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 2771	. . . . . . 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 3245	. . . . . 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 2830	. . . 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 2764	. . . . . . 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 3227	. . . . . 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 7429	. . . . . . . . . 10 ⊢ (𝑜 = 𝑣 → (𝑜 −s 𝐴) = (𝑣 −s 𝐴))
49	48	oveq1d 7437	. . . . . . . . 9 ⊢ (𝑜 = 𝑣 → ((𝑜 −s 𝐴) ·s (𝐵 −s 𝑥)) = ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑥)))
50	49	oveq2d 7438	. . . . . . . 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 2771	. . . . . . 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 7430	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝐵 −s 𝑥) = (𝐵 −s 𝑤))
53	52	oveq2d 7438	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑥)) = ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)))
54	53	oveq2d 7438	. . . . . . . 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 2771	. . . . . . 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 3245	. . . . . 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 2830	. . . 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 4164	. . 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 7434	. 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 2811	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)))})))

This proof depends on syntax axioms:   → wi 4   = wceq 1569  {cab 2738  ∃wrex 3086   ∪ cun 4148   class class class wbr 5102  (class class class)co 7422   <<s cslts 28150   |s ccuts 28152   +_s cadds 28352   −_s csubs 28413   ·_s cmuls 28499

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-1o 8481  df-2o 8482  df-nadd 8680  df-no 28007  df-lts 28008  df-bday 28009  df-les 28109  df-slts 28151  df-cuts 28153  df-0s 28200  df-made 28220  df-old 28221  df-left 28223  df-right 28224  df-norec 28331  df-norec2 28342  df-adds 28353  df-negs 28414  df-subs 28415  df-muls 28500

This theorem is used by:  remulscl  28895