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Theorem isrngohom 35348
Description: The predicate "is a ring homomorphism from 𝑅 to 𝑆". (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
rnghomval.1 𝐺 = (1st𝑅)
rnghomval.2 𝐻 = (2nd𝑅)
rnghomval.3 𝑋 = ran 𝐺
rnghomval.4 𝑈 = (GId‘𝐻)
rnghomval.5 𝐽 = (1st𝑆)
rnghomval.6 𝐾 = (2nd𝑆)
rnghomval.7 𝑌 = ran 𝐽
rnghomval.8 𝑉 = (GId‘𝐾)
Assertion
Ref Expression
isrngohom ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋𝑌 ∧ (𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑦,𝑌   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem isrngohom
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 rnghomval.1 . . . 4 𝐺 = (1st𝑅)
2 rnghomval.2 . . . 4 𝐻 = (2nd𝑅)
3 rnghomval.3 . . . 4 𝑋 = ran 𝐺
4 rnghomval.4 . . . 4 𝑈 = (GId‘𝐻)
5 rnghomval.5 . . . 4 𝐽 = (1st𝑆)
6 rnghomval.6 . . . 4 𝐾 = (2nd𝑆)
7 rnghomval.7 . . . 4 𝑌 = ran 𝐽
8 rnghomval.8 . . . 4 𝑉 = (GId‘𝐾)
91, 2, 3, 4, 5, 6, 7, 8rngohomval 35347 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngHom 𝑆) = {𝑓 ∈ (𝑌m 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))})
109eleq2d 2901 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))}))
115fvexi 6675 . . . . . . 7 𝐽 ∈ V
1211rnex 7612 . . . . . 6 ran 𝐽 ∈ V
137, 12eqeltri 2912 . . . . 5 𝑌 ∈ V
141fvexi 6675 . . . . . . 7 𝐺 ∈ V
1514rnex 7612 . . . . . 6 ran 𝐺 ∈ V
163, 15eqeltri 2912 . . . . 5 𝑋 ∈ V
1713, 16elmap 8431 . . . 4 (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌)
1817anbi1i 626 . . 3 ((𝐹 ∈ (𝑌m 𝑋) ∧ ((𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))) ↔ (𝐹:𝑋𝑌 ∧ ((𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))))
19 fveq1 6660 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑈) = (𝐹𝑈))
2019eqeq1d 2826 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑈) = 𝑉 ↔ (𝐹𝑈) = 𝑉))
21 fveq1 6660 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥𝐺𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
22 fveq1 6660 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
23 fveq1 6660 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
2422, 23oveq12d 7167 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)𝐽(𝑓𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
2521, 24eqeq12d 2840 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ↔ (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦))))
26 fveq1 6660 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥𝐻𝑦)) = (𝐹‘(𝑥𝐻𝑦)))
2722, 23oveq12d 7167 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)𝐾(𝑓𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))
2826, 27eqeq12d 2840 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦)) ↔ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))
2925, 28anbi12d 633 . . . . . 6 (𝑓 = 𝐹 → (((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))) ↔ ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))))
30292ralbidv 3194 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))) ↔ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))))
3120, 30anbi12d 633 . . . 4 (𝑓 = 𝐹 → (((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦)))) ↔ ((𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))))
3231elrab 3666 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))} ↔ (𝐹 ∈ (𝑌m 𝑋) ∧ ((𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))))
33 3anass 1092 . . 3 ((𝐹:𝑋𝑌 ∧ (𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))) ↔ (𝐹:𝑋𝑌 ∧ ((𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))))
3418, 32, 333bitr4i 306 . 2 (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))} ↔ (𝐹:𝑋𝑌 ∧ (𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))))
3510, 34syl6bb 290 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋𝑌 ∧ (𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  ran crn 5543  wf 6339  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  m cmap 8402  GIdcgi 28276  RingOpscrngo 35277   RngHom crnghom 35343
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-rngohom 35346
This theorem is referenced by:  rngohomf  35349  rngohom1  35351  rngohomadd  35352  rngohommul  35353  rngohomco  35357  rngoisocnv  35364
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