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Theorem ercpbl 16834
 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 (𝜑𝑉𝑊)
ercpbl.f 𝐹 = (𝑥𝑉 ↦ [𝑥] )
ercpbl.c ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)
ercpbl.e (𝜑 → ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷)))
Assertion
Ref Expression
ercpbl ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) → (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
StepHypRef Expression
1 ercpbl.e . . 3 (𝜑 → ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷)))
213ad2ant1 1130 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷)))
3 ercpbl.r . . . . 5 (𝜑 Er 𝑉)
433ad2ant1 1130 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → Er 𝑉)
5 ercpbl.v . . . . 5 (𝜑𝑉𝑊)
653ad2ant1 1130 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝑉𝑊)
7 ercpbl.f . . . 4 𝐹 = (𝑥𝑉 ↦ [𝑥] )
8 simp2l 1196 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐴𝑉)
94, 6, 7, 8ercpbllem 16833 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 𝐶))
10 simp2r 1197 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐵𝑉)
114, 6, 7, 10ercpbllem 16833 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹𝐵) = (𝐹𝐷) ↔ 𝐵 𝐷))
129, 11anbi12d 633 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 𝐶𝐵 𝐷)))
13 ercpbl.c . . . . 5 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)
1413caovclg 7331 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
15143adant3 1129 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
164, 6, 7, 15ercpbllem 16833 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷)) ↔ (𝐴 + 𝐵) (𝐶 + 𝐷)))
172, 12, 163imtr4d 297 1 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5034   ↦ cmpt 5114  ‘cfv 6332  (class class class)co 7145   Er wer 8287  [cec 8288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fv 6340  df-ov 7148  df-er 8290  df-ec 8292 This theorem is referenced by:  qusaddvallem  16836  qusaddflem  16837  qusgrp2  18230  qusring2  19387  quslmod  31023
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