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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngohom0 | Structured version Visualization version GIF version |
Description: A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.) |
Ref | Expression |
---|---|
rnghom0.1 | ⊢ 𝐺 = (1st ‘𝑅) |
rnghom0.2 | ⊢ 𝑍 = (GId‘𝐺) |
rnghom0.3 | ⊢ 𝐽 = (1st ‘𝑆) |
rnghom0.4 | ⊢ 𝑊 = (GId‘𝐽) |
Ref | Expression |
---|---|
rngohom0 | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑍) = 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghom0.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 35831 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | 2 | 3ad2ant1 1135 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐺 ∈ GrpOp) |
4 | rnghom0.3 | . . . 4 ⊢ 𝐽 = (1st ‘𝑆) | |
5 | 4 | rngogrpo 35831 | . . 3 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp) |
6 | 5 | 3ad2ant2 1136 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐽 ∈ GrpOp) |
7 | 1, 4 | rngogrphom 35892 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) |
8 | rnghom0.2 | . . 3 ⊢ 𝑍 = (GId‘𝐺) | |
9 | rnghom0.4 | . . 3 ⊢ 𝑊 = (GId‘𝐽) | |
10 | 8, 9 | ghomidOLD 35810 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹‘𝑍) = 𝑊) |
11 | 3, 6, 7, 10 | syl3anc 1373 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑍) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ‘cfv 6397 (class class class)co 7231 1st c1st 7777 GrpOpcgr 28594 GIdcgi 28595 GrpOpHom cghomOLD 35804 RingOpscrngo 35815 RngHom crnghom 35881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-1st 7779 df-2nd 7780 df-map 8530 df-grpo 28598 df-gid 28599 df-ablo 28650 df-ghomOLD 35805 df-rngo 35816 df-rngohom 35884 |
This theorem is referenced by: keridl 35953 |
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