Theorem ercpbl 16032

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

Hypotheses

Ref	Expression
ercpbl.r	⊢ (𝜑 → ∼ Er 𝑉)
ercpbl.v	⊢ (𝜑 → 𝑉 ∈ V)
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)

Proof of Theorem ercpbl

Step	Hyp	Ref	Expression
1		ercpbl.e	. . 3 ⊢ (𝜑 → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))
2	1	3ad2ant1 1075	. 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 1075	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ∼ Er 𝑉)
5		ercpbl.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ V)
6	5	3ad2ant1 1075	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑉 ∈ V)
7		ercpbl.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
8		simp2l 1080	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
9	4, 6, 7, 8	ercpbllem 16031	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐴) = (𝐹‘𝐶) ↔ 𝐴 ∼ 𝐶))
10		simp2r 1081	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
11	4, 6, 7, 10	ercpbllem 16031	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐷) ↔ 𝐵 ∼ 𝐷))
12	9, 11	anbi12d 743	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷)))
13		ercpbl.c	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)
14	13	caovclg 6724	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
15	14	3adant3 1074	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
16	4, 6, 7, 15	ercpbllem 16031	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷)) ↔ (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))
17	2, 12, 16	3imtr4d 282	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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   Er wer 7626  [cec 7627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-er 7629  df-ec 7631

This theorem is referenced by:  qusaddvallem  16034  qusaddflem  16035  qusgrp2  17356  qusring2  18443