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Mirrors > Home > MPE Home > Th. List > rhm1 | Structured version Visualization version GIF version |
Description: Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
Ref | Expression |
---|---|
rhm1.o | ⊢ 1 = (1r‘𝑅) |
rhm1.n | ⊢ 𝑁 = (1r‘𝑆) |
Ref | Expression |
---|---|
rhm1 | ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘ 1 ) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2728 | . . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
3 | 1, 2 | rhmmhm 20412 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) |
4 | eqid 2728 | . . . 4 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
5 | eqid 2728 | . . . 4 ⊢ (0g‘(mulGrp‘𝑆)) = (0g‘(mulGrp‘𝑆)) | |
6 | 4, 5 | mhm0 18745 | . . 3 ⊢ (𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)) → (𝐹‘(0g‘(mulGrp‘𝑅))) = (0g‘(mulGrp‘𝑆))) |
7 | 3, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘(0g‘(mulGrp‘𝑅))) = (0g‘(mulGrp‘𝑆))) |
8 | rhm1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
9 | 1, 8 | ringidval 20117 | . . 3 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
10 | 9 | fveq2i 6895 | . 2 ⊢ (𝐹‘ 1 ) = (𝐹‘(0g‘(mulGrp‘𝑅))) |
11 | rhm1.n | . . 3 ⊢ 𝑁 = (1r‘𝑆) | |
12 | 2, 11 | ringidval 20117 | . 2 ⊢ 𝑁 = (0g‘(mulGrp‘𝑆)) |
13 | 7, 10, 12 | 3eqtr4g 2793 | 1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘ 1 ) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7415 0gc0g 17415 MndHom cmhm 18732 mulGrpcmgp 20068 1rcur 20115 RingHom crh 20402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-0g 17417 df-mhm 18734 df-ghm 19162 df-mgp 20069 df-ur 20116 df-ring 20169 df-rhm 20405 |
This theorem is referenced by: rhmopp 20442 elrhmunit 20443 rhmunitinv 20444 nrhmzr 20468 srng1 20733 mulgrhm2 21398 zrh1 21432 mplind 22008 evlslem1 22022 cpmidgsumm2pm 22765 lgsqrlem1 27273 kerunit 33029 rhmquskerlem 33135 rhmqusnsg 33138 rhmpreimaprmidl 33162 fldhmf1 41556 ricdrng1 41755 rhmmpl 41777 |
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