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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 2821 | . . . . 5 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | rnghom1.1 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | eqid 2821 | . . . . 5 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
4 | rnghom1.2 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
5 | eqid 2821 | . . . . 5 ⊢ (1st ‘𝑆) = (1st ‘𝑆) | |
6 | rnghom1.3 | . . . . 5 ⊢ 𝐾 = (2nd ‘𝑆) | |
7 | eqid 2821 | . . . . 5 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆) | |
8 | rnghom1.4 | . . . . 5 ⊢ 𝑉 = (GId‘𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | isrngohom 35258 | . . . 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 479 | . . 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 1139 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) |
12 | 11 | 3impa 1106 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ran crn 5556 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 GIdcgi 28267 RingOpscrngo 35187 RngHom crnghom 35253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-rngohom 35256 |
This theorem is referenced by: rngohomco 35267 rngoisocnv 35274 |
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