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Theorem ercpbl 17596
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 1132 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷)))
3 ercpbl.r . . . . 5 (𝜑 Er 𝑉)
433ad2ant1 1132 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → Er 𝑉)
5 ercpbl.v . . . . 5 (𝜑𝑉𝑊)
653ad2ant1 1132 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝑉𝑊)
7 ercpbl.f . . . 4 𝐹 = (𝑥𝑉 ↦ [𝑥] )
8 simp2l 1198 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐴𝑉)
94, 6, 7, 8ercpbllem 17595 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 𝐶))
10 simp2r 1199 . . . 4 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐵𝑉)
114, 6, 7, 10ercpbllem 17595 . . 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 1131 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (𝐴 + 𝐵) ∈ 𝑉)
164, 6, 7, 15ercpbllem 17595 . 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 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cmpt 5231  cfv 6563  (class class class)co 7431   Er wer 8741  [cec 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-er 8744  df-ec 8746
This theorem is referenced by:  qusaddvallem  17598  qusaddflem  17599  qusgrp2  19089  qusrng  20198  qusring2  20348  quslmod  33366
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