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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 RngHom 𝑆)) → 𝐹:𝑋⟶𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghomf.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | eqid 2737 | . . . . 5 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | rnghomf.2 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | eqid 2737 | . . . . 5 ⊢ (GId‘(2nd ‘𝑅)) = (GId‘(2nd ‘𝑅)) | |
5 | rnghomf.3 | . . . . 5 ⊢ 𝐽 = (1st ‘𝑆) | |
6 | eqid 2737 | . . . . 5 ⊢ (2nd ‘𝑆) = (2nd ‘𝑆) | |
7 | rnghomf.4 | . . . . 5 ⊢ 𝑌 = ran 𝐽 | |
8 | eqid 2737 | . . . . 5 ⊢ (GId‘(2nd ‘𝑆)) = (GId‘(2nd ‘𝑆)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | isrngohom 36427 | . . . 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 478 | . . 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 1143 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋⟶𝑌) |
12 | 11 | 3impa 1111 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ran crn 5635 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 GIdcgi 29435 RingOpscrngo 36356 RngHom crnghom 36422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8768 df-rngohom 36425 |
This theorem is referenced by: rngohomcl 36429 rngogrphom 36433 rngohomco 36436 keridl 36494 |
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