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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 2610 | . . . . 5 ⊢ (1st ‘𝑅) = (1st ‘𝑅) | |
2 | rnghom1.1 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | eqid 2610 | . . . . 5 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅) | |
4 | rnghom1.2 | . . . . 5 ⊢ 𝑈 = (GId‘𝐻) | |
5 | eqid 2610 | . . . . 5 ⊢ (1st ‘𝑆) = (1st ‘𝑆) | |
6 | rnghom1.3 | . . . . 5 ⊢ 𝐾 = (2nd ‘𝑆) | |
7 | eqid 2610 | . . . . 5 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆) | |
8 | rnghom1.4 | . . . . 5 ⊢ 𝑉 = (GId‘𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | isrngohom 32934 | . . . 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 500 | . . 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 1067 | . 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) |
12 | 11 | 3impa 1251 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 GIdcgi 26728 RingOpscrngo 32863 RngHom crnghom 32929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-rngohom 32932 |
This theorem is referenced by: rngohomco 32943 rngoisocnv 32950 |
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