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Theorem rngohomval 32933
 Description: The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
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
rngohomval ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngHom 𝑆) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))})
Distinct variable groups:   𝑥,𝑓,𝑦   𝑓,𝐺   𝑓,𝐻   𝑓,𝐽   𝑓,𝑌,𝑦   𝑓,𝐾   𝑅,𝑓,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   𝑓,𝑋,𝑥,𝑦   𝑈,𝑓   𝑓,𝑉
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 rngohomval
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑠 = 𝑆)
21fveq2d 6107 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (1st𝑠) = (1st𝑆))
3 rnghomval.5 . . . . . . 7 𝐽 = (1st𝑆)
42, 3syl6eqr 2662 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (1st𝑠) = 𝐽)
54rneqd 5274 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (1st𝑠) = ran 𝐽)
6 rnghomval.7 . . . . 5 𝑌 = ran 𝐽
75, 6syl6eqr 2662 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (1st𝑠) = 𝑌)
8 simpl 472 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
98fveq2d 6107 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (1st𝑟) = (1st𝑅))
10 rnghomval.1 . . . . . . 7 𝐺 = (1st𝑅)
119, 10syl6eqr 2662 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (1st𝑟) = 𝐺)
1211rneqd 5274 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (1st𝑟) = ran 𝐺)
13 rnghomval.3 . . . . 5 𝑋 = ran 𝐺
1412, 13syl6eqr 2662 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (1st𝑟) = 𝑋)
157, 14oveq12d 6567 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (ran (1st𝑠) ↑𝑚 ran (1st𝑟)) = (𝑌𝑚 𝑋))
168fveq2d 6107 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → (2nd𝑟) = (2nd𝑅))
17 rnghomval.2 . . . . . . . . 9 𝐻 = (2nd𝑅)
1816, 17syl6eqr 2662 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (2nd𝑟) = 𝐻)
1918fveq2d 6107 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (GId‘(2nd𝑟)) = (GId‘𝐻))
20 rnghomval.4 . . . . . . 7 𝑈 = (GId‘𝐻)
2119, 20syl6eqr 2662 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (GId‘(2nd𝑟)) = 𝑈)
2221fveq2d 6107 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(GId‘(2nd𝑟))) = (𝑓𝑈))
231fveq2d 6107 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (2nd𝑠) = (2nd𝑆))
24 rnghomval.6 . . . . . . . 8 𝐾 = (2nd𝑆)
2523, 24syl6eqr 2662 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (2nd𝑠) = 𝐾)
2625fveq2d 6107 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (GId‘(2nd𝑠)) = (GId‘𝐾))
27 rnghomval.8 . . . . . 6 𝑉 = (GId‘𝐾)
2826, 27syl6eqr 2662 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (GId‘(2nd𝑠)) = 𝑉)
2922, 28eqeq12d 2625 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(GId‘(2nd𝑟))) = (GId‘(2nd𝑠)) ↔ (𝑓𝑈) = 𝑉))
3011oveqd 6566 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥(1st𝑟)𝑦) = (𝑥𝐺𝑦))
3130fveq2d 6107 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(𝑥(1st𝑟)𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
324oveqd 6566 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)))
3331, 32eqeq12d 2625 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(𝑥(1st𝑟)𝑦)) = ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) ↔ (𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦))))
3418oveqd 6566 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥(2nd𝑟)𝑦) = (𝑥𝐻𝑦))
3534fveq2d 6107 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(𝑥(2nd𝑟)𝑦)) = (𝑓‘(𝑥𝐻𝑦)))
3625oveqd 6566 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓𝑥)(2nd𝑠)(𝑓𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦)))
3735, 36eqeq12d 2625 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(𝑥(2nd𝑟)𝑦)) = ((𝑓𝑥)(2nd𝑠)(𝑓𝑦)) ↔ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))
3833, 37anbi12d 743 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (((𝑓‘(𝑥(1st𝑟)𝑦)) = ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(2nd𝑟)𝑦)) = ((𝑓𝑥)(2nd𝑠)(𝑓𝑦))) ↔ ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦)))))
3914, 38raleqbidv 3129 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (∀𝑦 ∈ ran (1st𝑟)((𝑓‘(𝑥(1st𝑟)𝑦)) = ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(2nd𝑟)𝑦)) = ((𝑓𝑥)(2nd𝑠)(𝑓𝑦))) ↔ ∀𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦)))))
4014, 39raleqbidv 3129 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (∀𝑥 ∈ ran (1st𝑟)∀𝑦 ∈ ran (1st𝑟)((𝑓‘(𝑥(1st𝑟)𝑦)) = ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(2nd𝑟)𝑦)) = ((𝑓𝑥)(2nd𝑠)(𝑓𝑦))) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦)))))
4129, 40anbi12d 743 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (((𝑓‘(GId‘(2nd𝑟))) = (GId‘(2nd𝑠)) ∧ ∀𝑥 ∈ ran (1st𝑟)∀𝑦 ∈ ran (1st𝑟)((𝑓‘(𝑥(1st𝑟)𝑦)) = ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(2nd𝑟)𝑦)) = ((𝑓𝑥)(2nd𝑠)(𝑓𝑦)))) ↔ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))))
4215, 41rabeqbidv 3168 . 2 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑓 ∈ (ran (1st𝑠) ↑𝑚 ran (1st𝑟)) ∣ ((𝑓‘(GId‘(2nd𝑟))) = (GId‘(2nd𝑠)) ∧ ∀𝑥 ∈ ran (1st𝑟)∀𝑦 ∈ ran (1st𝑟)((𝑓‘(𝑥(1st𝑟)𝑦)) = ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(2nd𝑟)𝑦)) = ((𝑓𝑥)(2nd𝑠)(𝑓𝑦))))} = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))})
43 df-rngohom 32932 . 2 RngHom = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (ran (1st𝑠) ↑𝑚 ran (1st𝑟)) ∣ ((𝑓‘(GId‘(2nd𝑟))) = (GId‘(2nd𝑠)) ∧ ∀𝑥 ∈ ran (1st𝑟)∀𝑦 ∈ ran (1st𝑟)((𝑓‘(𝑥(1st𝑟)𝑦)) = ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(2nd𝑟)𝑦)) = ((𝑓𝑥)(2nd𝑠)(𝑓𝑦))))})
44 ovex 6577 . . 3 (𝑌𝑚 𝑋) ∈ V
4544rabex 4740 . 2 {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))} ∈ V
4642, 43, 45ovmpt2a 6689 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngHom 𝑆) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  ran crn 5039  ‘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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-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-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rngohom 32932 This theorem is referenced by:  isrngohom  32934
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