Theorem ercpbl 12917

Description: Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)

Hypotheses

Ref	Expression
ercpbl.r	⊢ (𝜑 → ∼ Er 𝑉)
ercpbl.v	⊢ (𝜑 → 𝑉 ∈ 𝑊)
ercpbl.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
ercpbl.c	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)
ercpbl.e	⊢ (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))

Assertion

Ref	Expression
ercpbl	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))

Distinct variable groups:   𝑥, ∼   𝑎,𝑏,𝑥,𝐴   𝐵,𝑏,𝑥   𝑥,𝐶   𝑥,𝐷   𝑉,𝑎,𝑏,𝑥   + ,𝑎,𝑏,𝑥   𝜑,𝑎,𝑏,𝑥

Allowed substitution hints:   𝐵(𝑎)   𝐶(𝑎,𝑏)   𝐷(𝑎,𝑏)   ∼ (𝑎,𝑏)   𝐹(𝑥,𝑎,𝑏)   𝑊(𝑥,𝑎,𝑏)

Proof of Theorem ercpbl

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ercpbl.e	. . 3 ⊢ (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))
2	1	3ad2ant1 1020	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))
3		ercpbl.r	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑉)
4	3	3ad2ant1 1020	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ∼ Er 𝑉)
5		ercpbl.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
6	5	3ad2ant1 1020	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑉 ∈ 𝑊)
7		ercpbl.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
8		simp2l 1025	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
9		simp3l 1027	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
10	4, 6, 7, 8, 9	ercpbllemg 12916	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐴) = (𝐹‘𝐶) ↔ 𝐴 ∼ 𝐶))
11		simp2r 1026	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
12		simp3r 1028	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐷 ∈ 𝑉)
13	4, 6, 7, 11, 12	ercpbllemg 12916	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐷) ↔ 𝐵 ∼ 𝐷))
14	10, 13	anbi12d 473	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷)))
15		ercpbl.c	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)
16	15	caovclg 6073	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
17	16	3adant3 1019	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
18		simprl 529	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
19		simprr 531	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐷 ∈ 𝑉)
20	15	ralrimivva 2576	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑎 + 𝑏) ∈ 𝑉)
21		oveq1 5926	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → (𝑎 + 𝑏) = (𝑐 + 𝑏))
22	21	eleq1d 2262	. . . . . . . 8 ⊢ (𝑎 = 𝑐 → ((𝑎 + 𝑏) ∈ 𝑉 ↔ (𝑐 + 𝑏) ∈ 𝑉))
23		oveq2 5927	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (𝑐 + 𝑏) = (𝑐 + 𝑑))
24	23	eleq1d 2262	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((𝑐 + 𝑏) ∈ 𝑉 ↔ (𝑐 + 𝑑) ∈ 𝑉))
25	22, 24	cbvral2v 2739	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑎 + 𝑏) ∈ 𝑉 ↔ ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 + 𝑑) ∈ 𝑉)
26	20, 25	sylib 122	. . . . . 6 ⊢ (𝜑 → ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 + 𝑑) ∈ 𝑉)
27	26	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 + 𝑑) ∈ 𝑉)
28		ovrspc2v 5945	. . . . 5 ⊢ (((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) ∧ ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 + 𝑑) ∈ 𝑉) → (𝐶 + 𝐷) ∈ 𝑉)
29	18, 19, 27, 28	syl21anc 1248	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 + 𝐷) ∈ 𝑉)
30	29	3adant2 1018	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 + 𝐷) ∈ 𝑉)
31	4, 6, 7, 17, 30	ercpbllemg 12916	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷)) ↔ (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))
32	2, 14, 31	3imtr4d 203	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472   class class class wbr 4030   ↦ cmpt 4091  ‘cfv 5255  (class class class)co 5919   Er wer 6586  [cec 6587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-er 6589  df-ec 6591

This theorem is referenced by:  qusaddvallemg  12919  qusaddflemg  12920  qusgrp2  13186  qusrng  13457  qusring2  13565