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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngohomsub | Structured version Visualization version GIF version |
Description: Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.) |
Ref | Expression |
---|---|
rnghomsub.1 | ⊢ 𝐺 = (1st ‘𝑅) |
rnghomsub.2 | ⊢ 𝑋 = ran 𝐺 |
rnghomsub.3 | ⊢ 𝐻 = ( /𝑔 ‘𝐺) |
rnghomsub.4 | ⊢ 𝐽 = (1st ‘𝑆) |
rnghomsub.5 | ⊢ 𝐾 = ( /𝑔 ‘𝐽) |
Ref | Expression |
---|---|
rngohomsub | ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghomsub.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 35762 | . . . 4 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | 2 | 3ad2ant1 1135 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐺 ∈ GrpOp) |
4 | rnghomsub.4 | . . . . 5 ⊢ 𝐽 = (1st ‘𝑆) | |
5 | 4 | rngogrpo 35762 | . . . 4 ⊢ (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp) |
6 | 5 | 3ad2ant2 1136 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐽 ∈ GrpOp) |
7 | 1, 4 | rngogrphom 35823 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) |
8 | 3, 6, 7 | 3jca 1130 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽))) |
9 | rnghomsub.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
10 | rnghomsub.3 | . . 3 ⊢ 𝐻 = ( /𝑔 ‘𝐺) | |
11 | rnghomsub.5 | . . 3 ⊢ 𝐾 = ( /𝑔 ‘𝐽) | |
12 | 9, 10, 11 | ghomdiv 35744 | . 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))) |
13 | 8, 12 | sylan 583 | 1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ran crn 5541 ‘cfv 6369 (class class class)co 7202 1st c1st 7748 GrpOpcgr 28542 /𝑔 cgs 28545 GrpOpHom cghomOLD 35735 RingOpscrngo 35746 RngHom crnghom 35812 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
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 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-map 8499 df-grpo 28546 df-gid 28547 df-ginv 28548 df-gdiv 28549 df-ablo 28598 df-ghomOLD 35736 df-rngo 35747 df-rngohom 35815 |
This theorem is referenced by: (None) |
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