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Mirrors > Home > MPE Home > Th. List > rhmsubclem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for rhmsubc 20604. (Contributed by AV, 2-Mar-2020.) |
Ref | Expression |
---|---|
rngcrescrhm.u | β’ (π β π β π) |
rngcrescrhm.c | β’ πΆ = (RngCatβπ) |
rngcrescrhm.r | β’ (π β π = (Ring β© π)) |
rngcrescrhm.h | β’ π» = ( RingHom βΎ (π Γ π )) |
Ref | Expression |
---|---|
rhmsubclem3 | β’ ((π β§ π₯ β π ) β ((Idβ(RngCatβπ))βπ₯) β (π₯π»π₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcrescrhm.r | . . . . . 6 β’ (π β π = (Ring β© π)) | |
2 | 1 | eleq2d 2814 | . . . . 5 β’ (π β (π₯ β π β π₯ β (Ring β© π))) |
3 | elinel1 4191 | . . . . 5 β’ (π₯ β (Ring β© π) β π₯ β Ring) | |
4 | 2, 3 | biimtrdi 252 | . . . 4 β’ (π β (π₯ β π β π₯ β Ring)) |
5 | 4 | imp 406 | . . 3 β’ ((π β§ π₯ β π ) β π₯ β Ring) |
6 | eqid 2727 | . . . 4 β’ (Baseβπ₯) = (Baseβπ₯) | |
7 | 6 | idrhm 20411 | . . 3 β’ (π₯ β Ring β ( I βΎ (Baseβπ₯)) β (π₯ RingHom π₯)) |
8 | 5, 7 | syl 17 | . 2 β’ ((π β§ π₯ β π ) β ( I βΎ (Baseβπ₯)) β (π₯ RingHom π₯)) |
9 | rngcrescrhm.c | . . 3 β’ πΆ = (RngCatβπ) | |
10 | eqid 2727 | . . 3 β’ (BaseβπΆ) = (BaseβπΆ) | |
11 | 9 | eqcomi 2736 | . . . 4 β’ (RngCatβπ) = πΆ |
12 | 11 | fveq2i 6894 | . . 3 β’ (Idβ(RngCatβπ)) = (IdβπΆ) |
13 | rngcrescrhm.u | . . . 4 β’ (π β π β π) | |
14 | 13 | adantr 480 | . . 3 β’ ((π β§ π₯ β π ) β π β π) |
15 | incom 4197 | . . . . . 6 β’ (Ring β© π) = (π β© Ring) | |
16 | ringssrng 20204 | . . . . . . 7 β’ Ring β Rng | |
17 | sslin 4230 | . . . . . . 7 β’ (Ring β Rng β (π β© Ring) β (π β© Rng)) | |
18 | 16, 17 | mp1i 13 | . . . . . 6 β’ (π β (π β© Ring) β (π β© Rng)) |
19 | 15, 18 | eqsstrid 4026 | . . . . 5 β’ (π β (Ring β© π) β (π β© Rng)) |
20 | 9, 10, 13 | rngcbas 20536 | . . . . 5 β’ (π β (BaseβπΆ) = (π β© Rng)) |
21 | 19, 1, 20 | 3sstr4d 4025 | . . . 4 β’ (π β π β (BaseβπΆ)) |
22 | 21 | sselda 3978 | . . 3 β’ ((π β§ π₯ β π ) β π₯ β (BaseβπΆ)) |
23 | 9, 10, 12, 14, 22, 6 | rngcid 20550 | . 2 β’ ((π β§ π₯ β π ) β ((Idβ(RngCatβπ))βπ₯) = ( I βΎ (Baseβπ₯))) |
24 | rngcrescrhm.h | . . . 4 β’ π» = ( RingHom βΎ (π Γ π )) | |
25 | 13, 9, 1, 24 | rhmsubclem2 20601 | . . 3 β’ ((π β§ π₯ β π β§ π₯ β π ) β (π₯π»π₯) = (π₯ RingHom π₯)) |
26 | 25 | 3anidm23 1419 | . 2 β’ ((π β§ π₯ β π ) β (π₯π»π₯) = (π₯ RingHom π₯)) |
27 | 8, 23, 26 | 3eltr4d 2843 | 1 β’ ((π β§ π₯ β π ) β ((Idβ(RngCatβπ))βπ₯) β (π₯π»π₯)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β© cin 3943 β wss 3944 I cid 5569 Γ cxp 5670 βΎ cres 5674 βcfv 6542 (class class class)co 7414 Basecbs 17165 Idccid 17630 Rngcrng 20076 Ringcrg 20157 RingHom crh 20390 RngCatcrngc 20531 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-hom 17242 df-cco 17243 df-0g 17408 df-cat 17633 df-cid 17634 df-homf 17635 df-ssc 17778 df-resc 17779 df-subc 17780 df-estrc 18098 df-mgm 18585 df-mgmhm 18637 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-grp 18878 df-minusg 18879 df-ghm 19152 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-rnghm 20357 df-rhm 20393 df-rngc 20532 |
This theorem is referenced by: rhmsubc 20604 |
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