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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 44254 | . 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 18345 | . 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 1537 ∈ wcel 2114 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 GrpHom cghm 18338 RngHomo crngh 44241 |
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 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-map 8394 df-ghm 18339 df-abl 18892 df-rng0 44231 df-rnghomo 44243 |
This theorem is referenced by: rnghmf1o 44259 elrngchom 44324 rnghmsscmap2 44329 rnghmsscmap 44330 rnghmsubcsetclem2 44332 rngcsect 44336 rngcinv 44337 elrngchomALTV 44342 rngcinvALTV 44349 funcrngcsetc 44354 funcrngcsetcALT 44355 zrinitorngc 44356 zrtermorngc 44357 |
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