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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 RingOpsHom 𝑆)) → (𝐹‘𝑍) = 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghom0.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 37416 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | 2 | 3ad2ant1 1130 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐺 ∈ GrpOp) |
4 | rnghom0.3 | . . . 4 ⊢ 𝐽 = (1st ‘𝑆) | |
5 | 4 | rngogrpo 37416 | . . 3 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp) |
6 | 5 | 3ad2ant2 1131 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐽 ∈ GrpOp) |
7 | 1, 4 | rngogrphom 37477 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) |
8 | rnghom0.2 | . . 3 ⊢ 𝑍 = (GId‘𝐺) | |
9 | rnghom0.4 | . . 3 ⊢ 𝑊 = (GId‘𝐽) | |
10 | 8, 9 | ghomidOLD 37395 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) → (𝐹‘𝑍) = 𝑊) |
11 | 3, 6, 7, 10 | syl3anc 1368 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘𝑍) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 1st c1st 7997 GrpOpcgr 30319 GIdcgi 30320 GrpOpHom cghomOLD 37389 RingOpscrngo 37400 RingOpsHom crngohom 37466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 df-grpo 30323 df-gid 30324 df-ablo 30375 df-ghomOLD 37390 df-rngo 37401 df-rngohom 37469 |
This theorem is referenced by: keridl 37538 |
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