![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > rngohom1 | Structured version Visualization version GIF version |
Description: A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.) |
Ref | Expression |
---|---|
rnghom1.1 | ⊢ 𝐻 = (2nd ‘𝑅) |
rnghom1.2 | ⊢ 𝑈 = (GId‘𝐻) |
rnghom1.3 | ⊢ 𝐾 = (2nd ‘𝑆) |
rnghom1.4 | ⊢ 𝑉 = (GId‘𝐾) |
Ref | Expression |
---|---|
rngohom1 | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . 5 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | rnghom1.1 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | eqid 2798 | . . . . 5 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
4 | rnghom1.2 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
5 | eqid 2798 | . . . . 5 ⊢ (1st ‘𝑆) = (1st ‘𝑆) | |
6 | rnghom1.3 | . . . . 5 ⊢ 𝐾 = (2nd ‘𝑆) | |
7 | eqid 2798 | . . . . 5 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆) | |
8 | rnghom1.4 | . . . . 5 ⊢ 𝑉 = (GId‘𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | isrngohom 35403 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:ran (1st ‘𝑅)⟶ran (1st ‘𝑆) ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ ran (1st ‘𝑅)∀𝑦 ∈ ran (1st ‘𝑅)((𝐹‘(𝑥(1st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))) |
10 | 9 | biimpa 480 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st ‘𝑅)⟶ran (1st ‘𝑆) ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ ran (1st ‘𝑅)∀𝑦 ∈ ran (1st ‘𝑅)((𝐹‘(𝑥(1st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))) |
11 | 10 | simp2d 1140 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) |
12 | 11 | 3impa 1107 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 GIdcgi 28273 RingOpscrngo 35332 RngHom crnghom 35398 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
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 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-rngohom 35401 |
This theorem is referenced by: rngohomco 35412 rngoisocnv 35419 |
Copyright terms: Public domain | W3C validator |