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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 35203 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | 2 | 3ad2ant1 1129 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐺 ∈ GrpOp) |
4 | rnghom0.3 | . . . 4 ⊢ 𝐽 = (1st ‘𝑆) | |
5 | 4 | rngogrpo 35203 | . . 3 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp) |
6 | 5 | 3ad2ant2 1130 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐽 ∈ GrpOp) |
7 | 1, 4 | rngogrphom 35264 | . 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 35182 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹‘𝑍) = 𝑊) |
11 | 3, 6, 7, 10 | syl3anc 1367 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑍) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 GrpOpcgr 28266 GIdcgi 28267 GrpOpHom cghomOLD 35176 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-rep 5190 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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 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-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-grpo 28270 df-gid 28271 df-ablo 28322 df-ghomOLD 35177 df-rngo 35188 df-rngohom 35256 |
This theorem is referenced by: keridl 35325 |
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