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Mathbox for Jeff Madsen |
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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 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghomf.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | eqid 2726 | . . . . 5 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | rnghomf.2 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | eqid 2726 | . . . . 5 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
5 | rnghomf.3 | . . . . 5 ⊢ 𝐽 = (1st ‘𝑆) | |
6 | eqid 2726 | . . . . 5 ⊢ (2nd ‘𝑆) = (2nd ‘𝑆) | |
7 | rnghomf.4 | . . . . 5 ⊢ 𝑌 = ran 𝐽 | |
8 | eqid 2726 | . . . . 5 ⊢ (GId‘(2nd ‘𝑆)) = (GId‘(2nd ‘𝑆)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | isrngohom 37346 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsHom 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 476 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 1139 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌) |
12 | 11 | 3impa 1107 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ran crn 5670 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 1st c1st 7972 2nd c2nd 7973 GIdcgi 30252 RingOpscrngo 37275 RingOpsHom crngohom 37341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8824 df-rngohom 37344 |
This theorem is referenced by: rngohomcl 37348 rngogrphom 37352 rngohomco 37355 keridl 37413 |
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