Theorem mulsval2lem 28030

Description: Lemma for mulsval2 28031. Change bound variables in one of the cases. (Contributed by Scott Fenton, 8-Mar-2025.)

Assertion

Ref	Expression
mulsval2lem	⊢ {𝑎 ∣ ∃𝑝 ∈ 𝑋 ∃𝑞 ∈ 𝑌 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = {𝑏 ∣ ∃𝑟 ∈ 𝑋 ∃𝑠 ∈ 𝑌 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}

Distinct variable groups:   𝐴,𝑎,𝑏,𝑝,𝑞,𝑟   𝐴,𝑠   𝐵,𝑎,𝑏,𝑝,𝑞,𝑟   𝐵,𝑠   𝑋,𝑎,𝑏,𝑝,𝑟   𝑌,𝑎,𝑏,𝑝,𝑞,𝑟   𝑌,𝑠,𝑎,𝑏,𝑞,𝑟

Allowed substitution hints:   𝑋(𝑠,𝑞)

Proof of Theorem mulsval2lem

Step	Hyp	Ref	Expression
1		eqeq1 2729	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑏 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
2	1	2rexbidv 3210	. . 3 ⊢ (𝑎 = 𝑏 → (∃𝑝 ∈ 𝑋 ∃𝑞 ∈ 𝑌 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑝 ∈ 𝑋 ∃𝑞 ∈ 𝑌 𝑏 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
3		oveq1 7422	. . . . . . 7 ⊢ (𝑝 = 𝑟 → (𝑝 ·s 𝐵) = (𝑟 ·s 𝐵))
4	3	oveq1d 7430	. . . . . 6 ⊢ (𝑝 = 𝑟 → ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) = ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑞)))
5		oveq1 7422	. . . . . 6 ⊢ (𝑝 = 𝑟 → (𝑝 ·s 𝑞) = (𝑟 ·s 𝑞))
6	4, 5	oveq12d 7433	. . . . 5 ⊢ (𝑝 = 𝑟 → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑟 ·s 𝑞)))
7	6	eqeq2d 2736	. . . 4 ⊢ (𝑝 = 𝑟 → (𝑏 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑟 ·s 𝑞))))
8		oveq2 7423	. . . . . . 7 ⊢ (𝑞 = 𝑠 → (𝐴 ·s 𝑞) = (𝐴 ·s 𝑠))
9	8	oveq2d 7431	. . . . . 6 ⊢ (𝑞 = 𝑠 → ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑞)) = ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)))
10		oveq2 7423	. . . . . 6 ⊢ (𝑞 = 𝑠 → (𝑟 ·s 𝑞) = (𝑟 ·s 𝑠))
11	9, 10	oveq12d 7433	. . . . 5 ⊢ (𝑞 = 𝑠 → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑟 ·s 𝑞)) = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
12	11	eqeq2d 2736	. . . 4 ⊢ (𝑞 = 𝑠 → (𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑟 ·s 𝑞)) ↔ 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
13	7, 12	cbvrex2vw 3230	. . 3 ⊢ (∃𝑝 ∈ 𝑋 ∃𝑞 ∈ 𝑌 𝑏 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑟 ∈ 𝑋 ∃𝑠 ∈ 𝑌 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
14	2, 13	bitrdi 286	. 2 ⊢ (𝑎 = 𝑏 → (∃𝑝 ∈ 𝑋 ∃𝑞 ∈ 𝑌 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑟 ∈ 𝑋 ∃𝑠 ∈ 𝑌 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
15	14	cbvabv 2798	1 ⊢ {𝑎 ∣ ∃𝑝 ∈ 𝑋 ∃𝑞 ∈ 𝑌 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = {𝑏 ∣ ∃𝑟 ∈ 𝑋 ∃𝑠 ∈ 𝑌 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}

Syntax hints:   = wceq 1533  {cab 2702  ∃wrex 3060  (class class class)co 7415   +_s cadds 27892   -_s csubs 27949   ·_s cmuls 28026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-iota 6494  df-fv 6550  df-ov 7418

This theorem is referenced by:  mulsval2  28031  mulscut  28052  mulsunif  28070