Theorem mulsunif2 28211

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 28210	. 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 2739	. . . . . . 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 3220	. . . . . 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 7439	. . . . . . . . . 10 ⊢ (𝑖 = 𝑝 → (𝐴 -s 𝑖) = (𝐴 -s 𝑝))
9	8	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑖 = 𝑝 → ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)) = ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)))
10	9	oveq2d 7447	. . . . . . . 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 2746	. . . . . . 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 7439	. . . . . . . . . 10 ⊢ (𝑗 = 𝑞 → (𝐵 -s 𝑗) = (𝐵 -s 𝑞))
13	12	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑗 = 𝑞 → ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)) = ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))
14	13	oveq2d 7447	. . . . . . . 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 2746	. . . . . . 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 3240	. . . . . 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 287	. . . . 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 2810	. . . 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 2739	. . . . . . 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 3220	. . . . . 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 7438	. . . . . . . . . 10 ⊢ (𝑘 = 𝑟 → (𝑘 -s 𝐴) = (𝑟 -s 𝐴))
22	21	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑘 = 𝑟 → ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)) = ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)))
23	22	oveq2d 7447	. . . . . . . 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 2746	. . . . . . 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 7438	. . . . . . . . . 10 ⊢ (𝑙 = 𝑠 → (𝑙 -s 𝐵) = (𝑠 -s 𝐵))
26	25	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)) = ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))
27	26	oveq2d 7447	. . . . . . . 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 2746	. . . . . . 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 3240	. . . . . 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 287	. . . . 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 2810	. . . 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 4176	. . 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 2739	. . . . . . 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 3220	. . . . . 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 7439	. . . . . . . . . 10 ⊢ (𝑚 = 𝑡 → (𝐴 -s 𝑚) = (𝐴 -s 𝑡))
36	35	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑚 = 𝑡 → ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)) = ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)))
37	36	oveq2d 7447	. . . . . . . 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 2746	. . . . . . 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 7438	. . . . . . . . . 10 ⊢ (𝑛 = 𝑢 → (𝑛 -s 𝐵) = (𝑢 -s 𝐵))
40	39	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑛 = 𝑢 → ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)) = ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))
41	40	oveq2d 7447	. . . . . . . 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 2746	. . . . . . 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 3240	. . . . . 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 287	. . . . 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 2810	. . . 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 2739	. . . . . . 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 3220	. . . . . 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 7438	. . . . . . . . . 10 ⊢ (𝑜 = 𝑣 → (𝑜 -s 𝐴) = (𝑣 -s 𝐴))
49	48	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑜 = 𝑣 → ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)) = ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)))
50	49	oveq2d 7447	. . . . . . . 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 2746	. . . . . . 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 7439	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝐵 -s 𝑥) = (𝐵 -s 𝑤))
53	52	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)) = ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))
54	53	oveq2d 7447	. . . . . . . 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 2746	. . . . . . 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 3240	. . . . . 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 287	. . . . 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 2810	. . . 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 4176	. . 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 7443	. 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 2791	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 1537  {cab 2712  ∃wrex 3068   ∪ cun 3961   class class class wbr 5148  (class class class)co 7431   <<s csslt 27840   |s cscut 27842   +_s cadds 28007   -_s csubs 28067   ·_s cmuls 28147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148

This theorem is referenced by:  remulscl  28449