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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnghmf | Structured version Visualization version GIF version |
Description: A ring homomorphism is a function. (Contributed by AV, 23-Feb-2020.) |
Ref | Expression |
---|---|
rnghmf.b | ⊢ 𝐵 = (Base‘𝑅) |
rnghmf.c | ⊢ 𝐶 = (Base‘𝑆) |
Ref | Expression |
---|---|
rnghmf | ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹:𝐵⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghmghm 42759 | . 2 ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
2 | rnghmf.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rnghmf.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
4 | 2, 3 | ghmf 18022 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐵⟶𝐶) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 GrpHom cghm 18015 RngHomo crngh 42746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-ghm 18016 df-abl 18556 df-rng0 42736 df-rnghomo 42748 |
This theorem is referenced by: rnghmf1o 42764 elrngchom 42829 rnghmsscmap2 42834 rnghmsscmap 42835 rnghmsubcsetclem2 42837 rngcsect 42841 rngcinv 42842 elrngchomALTV 42847 rngcinvALTV 42854 funcrngcsetc 42859 funcrngcsetcALT 42860 zrinitorngc 42861 zrtermorngc 42862 |
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