Theorem ercpbl 13598

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 1045	. 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 1045	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ∼ Er 𝑉)
5		ercpbl.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
6	5	3ad2ant1 1045	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑉 ∈ 𝑊)
7		ercpbl.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
8		simp2l 1050	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
9		simp3l 1052	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
10	4, 6, 7, 8, 9	ercpbllemg 13597	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐴) = (𝐹‘𝐶) ↔ 𝐴 ∼ 𝐶))
11		simp2r 1051	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
12		simp3r 1053	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐷 ∈ 𝑉)
13	4, 6, 7, 11, 12	ercpbllemg 13597	. . 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 6215	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
17	16	3adant3 1044	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
18		simprl 531	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
19		simprr 533	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐷 ∈ 𝑉)
20	15	ralrimivva 2626	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑎 + 𝑏) ∈ 𝑉)
21		oveq1 6065	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → (𝑎 + 𝑏) = (𝑐 + 𝑏))
22	21	eleq1d 2303	. . . . . . . 8 ⊢ (𝑎 = 𝑐 → ((𝑎 + 𝑏) ∈ 𝑉 ↔ (𝑐 + 𝑏) ∈ 𝑉))
23		oveq2 6066	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (𝑐 + 𝑏) = (𝑐 + 𝑑))
24	23	eleq1d 2303	. . . . . . . 8 ⊢ (𝑏 = 𝑑 → ((𝑐 + 𝑏) ∈ 𝑉 ↔ (𝑐 + 𝑑) ∈ 𝑉))
25	22, 24	cbvral2v 2793	. . . . . . 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 6084	. . . . 5 ⊢ (((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) ∧ ∀𝑐 ∈ 𝑉 ∀𝑑 ∈ 𝑉 (𝑐 + 𝑑) ∈ 𝑉) → (𝐶 + 𝐷) ∈ 𝑉)
29	18, 19, 27, 28	syl21anc 1273	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 + 𝐷) ∈ 𝑉)
30	29	3adant2 1043	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 + 𝐷) ∈ 𝑉)
31	4, 6, 7, 17, 30	ercpbllemg 13597	. 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 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522   class class class wbr 4114   ↦ cmpt 4176  ‘cfv 5357  (class class class)co 6058   Er wer 6777  [cec 6778

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-er 6780  df-ec 6782

This theorem is referenced by:  qusaddvallemg  13600  qusaddflemg  13601  qusgrp2  13869  qusrng  14200  qusring2  14312