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Mathbox for Jeff Madsen |
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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 2778 | . . . . 5 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | rnghom1.1 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | eqid 2778 | . . . . 5 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
4 | rnghom1.2 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
5 | eqid 2778 | . . . . 5 ⊢ (1st ‘𝑆) = (1st ‘𝑆) | |
6 | rnghom1.3 | . . . . 5 ⊢ 𝐾 = (2nd ‘𝑆) | |
7 | eqid 2778 | . . . . 5 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆) | |
8 | rnghom1.4 | . . . . 5 ⊢ 𝑉 = (GId‘𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | isrngohom 34388 | . . . 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 470 | . . 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 1134 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) |
12 | 11 | 3impa 1097 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ran crn 5356 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 1st c1st 7443 2nd c2nd 7444 GIdcgi 27917 RingOpscrngo 34317 RngHom crnghom 34383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-map 8142 df-rngohom 34386 |
This theorem is referenced by: rngohomco 34397 rngoisocnv 34404 |
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