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Theorem ercpbl 17594
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 1134 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷)))
3 ercpbl.r . . . . 5 (𝜑 Er 𝑉)
433ad2ant1 1134 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → Er 𝑉)
5 ercpbl.v . . . . 5 (𝜑𝑉𝑊)
653ad2ant1 1134 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝑉𝑊)
7 ercpbl.f . . . 4 𝐹 = (𝑥𝑉 ↦ [𝑥] )
8 simp2l 1200 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐴𝑉)
94, 6, 7, 8ercpbllem 17593 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 𝐶))
10 simp2r 1201 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐵𝑉)
114, 6, 7, 10ercpbllem 17593 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹𝐵) = (𝐹𝐷) ↔ 𝐵 𝐷))
129, 11anbi12d 632 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 𝐶𝐵 𝐷)))
13 ercpbl.c . . . . 5 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎 + 𝑏) ∈ 𝑉)
1413caovclg 7625 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
15143adant3 1133 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
164, 6, 7, 15ercpbllem 17593 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷)) ↔ (𝐴 + 𝐵) (𝐶 + 𝐷)))
172, 12, 163imtr4d 294 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 395  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431   Er wer 8742  [cec 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-er 8745  df-ec 8747
This theorem is referenced by:  qusaddvallem  17596  qusaddflem  17597  qusgrp2  19076  qusrng  20177  qusring2  20331  quslmod  33386
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