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Theorem isrngohom 32934
 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 32933 . . 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 2673 . 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))))}))
11 fvex 6113 . . . . . . . 8 (1st𝑆) ∈ V
125, 11eqeltri 2684 . . . . . . 7 𝐽 ∈ V
1312rnex 6992 . . . . . 6 ran 𝐽 ∈ V
147, 13eqeltri 2684 . . . . 5 𝑌 ∈ V
15 fvex 6113 . . . . . . . 8 (1st𝑅) ∈ V
161, 15eqeltri 2684 . . . . . . 7 𝐺 ∈ V
1716rnex 6992 . . . . . 6 ran 𝐺 ∈ V
183, 17eqeltri 2684 . . . . 5 𝑋 ∈ V
1914, 18elmap 7772 . . . 4 (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌)
2019anbi1i 727 . . 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))))))
21 fveq1 6102 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑈) = (𝐹𝑈))
2221eqeq1d 2612 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑈) = 𝑉 ↔ (𝐹𝑈) = 𝑉))
23 fveq1 6102 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥𝐺𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
24 fveq1 6102 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
25 fveq1 6102 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
2624, 25oveq12d 6567 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)𝐽(𝑓𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
2723, 26eqeq12d 2625 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ↔ (𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦))))
28 fveq1 6102 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥𝐻𝑦)) = (𝐹‘(𝑥𝐻𝑦)))
2924, 25oveq12d 6567 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)𝐾(𝑓𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))
3028, 29eqeq12d 2625 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦)) ↔ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))
3127, 30anbi12d 743 . . . . . 6 (𝑓 = 𝐹 → (((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))) ↔ ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦)))))
32312ralbidv 2972 . . . . 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)))))
3322, 32anbi12d 743 . . . 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))))))
3433elrab 3331 . . 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))))))
35 3anass 1035 . . 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))))))
3620, 34, 353bitr4i 291 . 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)))))
3710, 36syl6bb 275 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  GIdcgi 26728  RingOpscrngo 32863   RngHom crnghom 32929 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-rngohom 32932 This theorem is referenced by:  rngohomf  32935  rngohom1  32937  rngohomadd  32938  rngohommul  32939  rngohomco  32943  rngoisocnv  32950
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