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Theorem rngohomval 35401
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 488 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑠 = 𝑆)
21fveq2d 6653 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (1st𝑠) = (1st𝑆))
3 rnghomval.5 . . . . . . 7 𝐽 = (1st𝑆)
42, 3eqtr4di 2854 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (1st𝑠) = 𝐽)
54rneqd 5776 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (1st𝑠) = ran 𝐽)
6 rnghomval.7 . . . . 5 𝑌 = ran 𝐽
75, 6eqtr4di 2854 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (1st𝑠) = 𝑌)
8 simpl 486 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
98fveq2d 6653 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (1st𝑟) = (1st𝑅))
10 rnghomval.1 . . . . . . 7 𝐺 = (1st𝑅)
119, 10eqtr4di 2854 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (1st𝑟) = 𝐺)
1211rneqd 5776 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (1st𝑟) = ran 𝐺)
13 rnghomval.3 . . . . 5 𝑋 = ran 𝐺
1412, 13eqtr4di 2854 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (1st𝑟) = 𝑋)
157, 14oveq12d 7157 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (ran (1st𝑠) ↑m ran (1st𝑟)) = (𝑌m 𝑋))
168fveq2d 6653 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → (2nd𝑟) = (2nd𝑅))
17 rnghomval.2 . . . . . . . . 9 𝐻 = (2nd𝑅)
1816, 17eqtr4di 2854 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (2nd𝑟) = 𝐻)
1918fveq2d 6653 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (GId‘(2nd𝑟)) = (GId‘𝐻))
20 rnghomval.4 . . . . . . 7 𝑈 = (GId‘𝐻)
2119, 20eqtr4di 2854 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (GId‘(2nd𝑟)) = 𝑈)
2221fveq2d 6653 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(GId‘(2nd𝑟))) = (𝑓𝑈))
231fveq2d 6653 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (2nd𝑠) = (2nd𝑆))
24 rnghomval.6 . . . . . . . 8 𝐾 = (2nd𝑆)
2523, 24eqtr4di 2854 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (2nd𝑠) = 𝐾)
2625fveq2d 6653 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (GId‘(2nd𝑠)) = (GId‘𝐾))
27 rnghomval.8 . . . . . 6 𝑉 = (GId‘𝐾)
2826, 27eqtr4di 2854 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (GId‘(2nd𝑠)) = 𝑉)
2922, 28eqeq12d 2817 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(GId‘(2nd𝑟))) = (GId‘(2nd𝑠)) ↔ (𝑓𝑈) = 𝑉))
3011oveqd 7156 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥(1st𝑟)𝑦) = (𝑥𝐺𝑦))
3130fveq2d 6653 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(𝑥(1st𝑟)𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
324oveqd 7156 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)))
3331, 32eqeq12d 2817 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(𝑥(1st𝑟)𝑦)) = ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) ↔ (𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦))))
3418oveqd 7156 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥(2nd𝑟)𝑦) = (𝑥𝐻𝑦))
3534fveq2d 6653 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(𝑥(2nd𝑟)𝑦)) = (𝑓‘(𝑥𝐻𝑦)))
3625oveqd 7156 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓𝑥)(2nd𝑠)(𝑓𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦)))
3735, 36eqeq12d 2817 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(𝑥(2nd𝑟)𝑦)) = ((𝑓𝑥)(2nd𝑠)(𝑓𝑦)) ↔ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))
3833, 37anbi12d 633 . . . . . 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 3357 . . . . 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 3357 . . . 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 633 . . 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 3436 . 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 35400 . 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 7172 . . 3 (𝑌m 𝑋) ∈ V
4544rabex 5202 . 2 {𝑓 ∈ (𝑌m 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))} ∈ V
4642, 43, 45ovmpoa 7288 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 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  ran crn 5524  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  m cmap 8393  GIdcgi 28277  RingOpscrngo 35331   RngHom crnghom 35397
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-rngohom 35400
This theorem is referenced by:  isrngohom  35402
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