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Mirrors > Home > MPE Home > Th. List > smadiadetlem3lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for smadiadetlem3 22557. (Contributed by AV, 12-Jan-2019.) |
Ref | Expression |
---|---|
marep01ma.a | β’ π΄ = (π Mat π ) |
marep01ma.b | β’ π΅ = (Baseβπ΄) |
marep01ma.r | β’ π β CRing |
marep01ma.0 | β’ 0 = (0gβπ ) |
marep01ma.1 | β’ 1 = (1rβπ ) |
smadiadetlem.p | β’ π = (Baseβ(SymGrpβπ)) |
smadiadetlem.g | β’ πΊ = (mulGrpβπ ) |
madetminlem.y | β’ π = (β€RHomβπ ) |
madetminlem.s | β’ π = (pmSgnβπ) |
madetminlem.t | β’ Β· = (.rβπ ) |
smadiadetlem.w | β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) |
smadiadetlem.z | β’ π = (pmSgnβ(π β {πΎ})) |
Ref | Expression |
---|---|
smadiadetlem3lem2 | β’ ((π β π΅ β§ πΎ β π) β ran (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))) β ((Cntzβπ )βran (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | marep01ma.r | . . 3 β’ π β CRing | |
2 | crngring 20176 | . . 3 β’ (π β CRing β π β Ring) | |
3 | ringcmn 20207 | . . 3 β’ (π β Ring β π β CMnd) | |
4 | 1, 2, 3 | mp2b 10 | . 2 β’ π β CMnd |
5 | marep01ma.a | . . . . 5 β’ π΄ = (π Mat π ) | |
6 | marep01ma.b | . . . . 5 β’ π΅ = (Baseβπ΄) | |
7 | marep01ma.0 | . . . . 5 β’ 0 = (0gβπ ) | |
8 | marep01ma.1 | . . . . 5 β’ 1 = (1rβπ ) | |
9 | smadiadetlem.p | . . . . 5 β’ π = (Baseβ(SymGrpβπ)) | |
10 | smadiadetlem.g | . . . . 5 β’ πΊ = (mulGrpβπ ) | |
11 | madetminlem.y | . . . . 5 β’ π = (β€RHomβπ ) | |
12 | madetminlem.s | . . . . 5 β’ π = (pmSgnβπ) | |
13 | madetminlem.t | . . . . 5 β’ Β· = (.rβπ ) | |
14 | smadiadetlem.w | . . . . 5 β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) | |
15 | smadiadetlem.z | . . . . 5 β’ π = (pmSgnβ(π β {πΎ})) | |
16 | 5, 6, 1, 7, 8, 9, 10, 11, 12, 13, 14, 15 | smadiadetlem3lem0 22554 | . . . 4 β’ (((π β π΅ β§ πΎ β π) β§ π β π) β (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ))))) β (Baseβπ )) |
17 | 16 | ralrimiva 3141 | . . 3 β’ ((π β π΅ β§ πΎ β π) β βπ β π (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ))))) β (Baseβπ )) |
18 | eqid 2727 | . . . 4 β’ (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))) = (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))) | |
19 | 18 | rnmptss 7127 | . . 3 β’ (βπ β π (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ))))) β (Baseβπ ) β ran (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))) β (Baseβπ )) |
20 | 17, 19 | syl 17 | . 2 β’ ((π β π΅ β§ πΎ β π) β ran (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))) β (Baseβπ )) |
21 | eqid 2727 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
22 | eqid 2727 | . . 3 β’ (Cntzβπ ) = (Cntzβπ ) | |
23 | 21, 22 | cntzcmnss 19787 | . 2 β’ ((π β CMnd β§ ran (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))) β (Baseβπ )) β ran (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))) β ((Cntzβπ )βran (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))))) |
24 | 4, 20, 23 | sylancr 586 | 1 β’ ((π β π΅ β§ πΎ β π) β ran (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))) β ((Cntzβπ )βran (π β π β¦ (((π β π)βπ)(.rβπ )(πΊ Ξ£g (π β (π β {πΎ}) β¦ (π(π β (π β {πΎ}), π β (π β {πΎ}) β¦ (πππ))(πβπ)))))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3056 β cdif 3941 β wss 3944 {csn 4624 β¦ cmpt 5225 ran crn 5673 β ccom 5676 βcfv 6542 (class class class)co 7414 β cmpo 7416 Basecbs 17171 .rcmulr 17225 0gc0g 17412 Ξ£g cgsu 17413 Cntzccntz 19257 SymGrpcsymg 19312 pmSgncpsgn 19435 CMndccmn 19726 mulGrpcmgp 20065 1rcur 20112 Ringcrg 20164 CRingccrg 20165 β€RHomczrh 21412 Mat cmat 22294 |
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 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-addf 11209 ax-mulf 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-xor 1506 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-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 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-se 5628 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-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8838 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-xnn0 12567 df-z 12581 df-dec 12700 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-word 14489 df-lsw 14537 df-concat 14545 df-s1 14570 df-substr 14615 df-pfx 14645 df-splice 14724 df-reverse 14733 df-s2 14823 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-efmnd 18812 df-grp 18884 df-minusg 18885 df-mulg 19015 df-subg 19069 df-ghm 19159 df-gim 19204 df-cntz 19259 df-oppg 19288 df-symg 19313 df-pmtr 19388 df-psgn 19437 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-sra 21047 df-rgmod 21048 df-cnfld 21267 df-zring 21360 df-zrh 21416 df-dsmm 21653 df-frlm 21668 df-mat 22295 |
This theorem is referenced by: smadiadetlem3 22557 |
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