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Theorem isrngohom 37952
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 RingOpsHom 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 37951 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsHom 𝑆) = {𝑓 ∈ (𝑌m 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))})
109eleq2d 2825 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))}))
115fvexi 6921 . . . . . . 7 𝐽 ∈ V
1211rnex 7933 . . . . . 6 ran 𝐽 ∈ V
137, 12eqeltri 2835 . . . . 5 𝑌 ∈ V
141fvexi 6921 . . . . . . 7 𝐺 ∈ V
1514rnex 7933 . . . . . 6 ran 𝐺 ∈ V
163, 15eqeltri 2835 . . . . 5 𝑋 ∈ V
1713, 16elmap 8910 . . . 4 (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌)
1817anbi1i 624 . . 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 6906 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑈) = (𝐹𝑈))
2019eqeq1d 2737 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑈) = 𝑉 ↔ (𝐹𝑈) = 𝑉))
21 fveq1 6906 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥𝐺𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
22 fveq1 6906 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
23 fveq1 6906 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
2422, 23oveq12d 7449 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)𝐽(𝑓𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
2521, 24eqeq12d 2751 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ↔ (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦))))
26 fveq1 6906 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥𝐻𝑦)) = (𝐹‘(𝑥𝐻𝑦)))
2722, 23oveq12d 7449 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)𝐾(𝑓𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))
2826, 27eqeq12d 2751 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦)) ↔ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))
2925, 28anbi12d 632 . . . . . 6 (𝑓 = 𝐹 → (((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))) ↔ ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))))
30292ralbidv 3219 . . . . 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 632 . . . 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 3695 . . 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 1094 . . 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 303 . 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, 34bitrdi 287 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  GIdcgi 30519  RingOpscrngo 37881   RingOpsHom crngohom 37947
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-pow 5371  ax-pr 5438  ax-un 7754
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-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-rngohom 37950
This theorem is referenced by:  rngohomf  37953  rngohom1  37955  rngohomadd  37956  rngohommul  37957  rngohomco  37961  rngoisocnv  37968
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