Theorem isrngohom 37137

Description: The predicate "is a ring homomorphism from 𝑅 to 𝑆". (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
rnghomval.1	⊢ 𝐺 = (1st ‘𝑅)
rnghomval.2	⊢ 𝐻 = (2nd ‘𝑅)
rnghomval.3	⊢ 𝑋 = ran 𝐺
rnghomval.4	⊢ 𝑈 = (GId‘𝐻)
rnghomval.5	⊢ 𝐽 = (1st ‘𝑆)
rnghomval.6	⊢ 𝐾 = (2nd ‘𝑆)
rnghomval.7	⊢ 𝑌 = ran 𝐽
rnghomval.8	⊢ 𝑉 = (GId‘𝐾)

Assertion

Ref	Expression
isrngohom	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))

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.

Step	Hyp	Ref	Expression
1		rnghomval.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		rnghomval.2	. . . 4 ⊢ 𝐻 = (2nd ‘𝑅)
3		rnghomval.3	. . . 4 ⊢ 𝑋 = ran 𝐺
4		rnghomval.4	. . . 4 ⊢ 𝑈 = (GId‘𝐻)
5		rnghomval.5	. . . 4 ⊢ 𝐽 = (1st ‘𝑆)
6		rnghomval.6	. . . 4 ⊢ 𝐾 = (2nd ‘𝑆)
7		rnghomval.7	. . . 4 ⊢ 𝑌 = ran 𝐽
8		rnghomval.8	. . . 4 ⊢ 𝑉 = (GId‘𝐾)
9	1, 2, 3, 4, 5, 6, 7, 8	rngohomval 37136	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RingOpsHom 𝑆) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ((𝑓‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦))))})
10	9	eleq2d 2818	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ((𝑓‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦))))}))
11	5	fvexi 6905	. . . . . . 7 ⊢ 𝐽 ∈ V
12	11	rnex 7907	. . . . . 6 ⊢ ran 𝐽 ∈ V
13	7, 12	eqeltri 2828	. . . . 5 ⊢ 𝑌 ∈ V
14	1	fvexi 6905	. . . . . . 7 ⊢ 𝐺 ∈ V
15	14	rnex 7907	. . . . . 6 ⊢ ran 𝐺 ∈ V
16	3, 15	eqeltri 2828	. . . . 5 ⊢ 𝑋 ∈ V
17	13, 16	elmap 8869	. . . 4 ⊢ (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌)
18	17	anbi1i 623	. . 3 ⊢ ((𝐹 ∈ (𝑌 ↑m 𝑋) ∧ ((𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))) ↔ (𝐹:𝑋⟶𝑌 ∧ ((𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))
19		fveq1 6890	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑈) = (𝐹‘𝑈))
20	19	eqeq1d 2733	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑈) = 𝑉 ↔ (𝐹‘𝑈) = 𝑉))
21		fveq1 6890	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥𝐺𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
22		fveq1 6890	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
23		fveq1 6890	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
24	22, 23	oveq12d 7430	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
25	21, 24	eqeq12d 2747	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ↔ (𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
26		fveq1 6890	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥𝐻𝑦)) = (𝐹‘(𝑥𝐻𝑦)))
27	22, 23	oveq12d 7430	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥)𝐾(𝑓‘𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
28	26, 27	eqeq12d 2747	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦)) ↔ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))
29	25, 28	anbi12d 630	. . . . . 6 ⊢ (𝑓 = 𝐹 → (((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦))) ↔ ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))
30	29	2ralbidv 3217	. . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))
31	20, 30	anbi12d 630	. . . 4 ⊢ (𝑓 = 𝐹 → (((𝑓‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦)))) ↔ ((𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))
32	31	elrab 3683	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ((𝑓‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦))))} ↔ (𝐹 ∈ (𝑌 ↑m 𝑋) ∧ ((𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))
33		3anass 1094	. . 3 ⊢ ((𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))) ↔ (𝐹:𝑋⟶𝑌 ∧ ((𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))
34	18, 32, 33	3bitr4i 303	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ((𝑓‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦))))} ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))
35	10, 34	bitrdi 287	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060  {crab 3431  Vcvv 3473  ran crn 5677  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412  1^st c1st 7977  2^nd c2nd 7978   ↑_m cmap 8824  GIdcgi 30011  RingOpscrngo 37066   RingOpsHom crngohom 37132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8826  df-rngohom 37135

This theorem is referenced by:  rngohomf  37138  rngohom1  37140  rngohomadd  37141  rngohommul  37142  rngohomco  37146  rngoisocnv  37153