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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngohomf | Structured version Visualization version GIF version |
Description: A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.) |
Ref | Expression |
---|---|
rnghomf.1 | ⊢ 𝐺 = (1st ‘𝑅) |
rnghomf.2 | ⊢ 𝑋 = ran 𝐺 |
rnghomf.3 | ⊢ 𝐽 = (1st ‘𝑆) |
rnghomf.4 | ⊢ 𝑌 = ran 𝐽 |
Ref | Expression |
---|---|
rngohomf | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋⟶𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghomf.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | eqid 2818 | . . . . 5 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | rnghomf.2 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | eqid 2818 | . . . . 5 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
5 | rnghomf.3 | . . . . 5 ⊢ 𝐽 = (1st ‘𝑆) | |
6 | eqid 2818 | . . . . 5 ⊢ (2nd ‘𝑆) = (2nd ‘𝑆) | |
7 | rnghomf.4 | . . . . 5 ⊢ 𝑌 = ran 𝐽 | |
8 | eqid 2818 | . . . . 5 ⊢ (GId‘(2nd ‘𝑆)) = (GId‘(2nd ‘𝑆)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | isrngohom 35124 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘(GId‘(2nd ‘𝑅))) = (GId‘(2nd ‘𝑆)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑦)))))) |
10 | 9 | biimpa 477 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋⟶𝑌 ∧ (𝐹‘(GId‘(2nd ‘𝑅))) = (GId‘(2nd ‘𝑆)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥(2nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2nd ‘𝑆)(𝐹‘𝑦))))) |
11 | 10 | simp1d 1134 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋⟶𝑌) |
12 | 11 | 3impa 1102 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ran crn 5549 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 GIdcgi 28194 RingOpscrngo 35053 RngHom crnghom 35119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-rngohom 35122 |
This theorem is referenced by: rngohomcl 35126 rngogrphom 35130 rngohomco 35133 keridl 35191 |
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