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Mirrors > Home > MPE Home > Th. List > smadiadetlem3lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for smadiadetlem3 21419. (Contributed by AV, 12-Jan-2019.) |
Ref | Expression |
---|---|
marep01ma.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
marep01ma.b | ⊢ 𝐵 = (Base‘𝐴) |
marep01ma.r | ⊢ 𝑅 ∈ CRing |
marep01ma.0 | ⊢ 0 = (0g‘𝑅) |
marep01ma.1 | ⊢ 1 = (1r‘𝑅) |
smadiadetlem.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
smadiadetlem.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
madetminlem.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
madetminlem.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
madetminlem.t | ⊢ · = (.r‘𝑅) |
smadiadetlem.w | ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) |
smadiadetlem.z | ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾})) |
Ref | Expression |
---|---|
smadiadetlem3lem1 | ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛)))))):𝑊⟶(Base‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | marep01ma.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | marep01ma.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
3 | marep01ma.r | . . 3 ⊢ 𝑅 ∈ CRing | |
4 | marep01ma.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
5 | marep01ma.1 | . . 3 ⊢ 1 = (1r‘𝑅) | |
6 | smadiadetlem.p | . . 3 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
7 | smadiadetlem.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
8 | madetminlem.y | . . 3 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
9 | madetminlem.s | . . 3 ⊢ 𝑆 = (pmSgn‘𝑁) | |
10 | madetminlem.t | . . 3 ⊢ · = (.r‘𝑅) | |
11 | smadiadetlem.w | . . 3 ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) | |
12 | smadiadetlem.z | . . 3 ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾})) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | smadiadetlem3lem0 21416 | . 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ 𝑊) → (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))) ∈ (Base‘𝑅)) |
14 | 13 | fmpttd 6889 | 1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛)))))):𝑊⟶(Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∖ cdif 3840 {csn 4516 ↦ cmpt 5110 ∘ ccom 5529 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 Basecbs 16586 .rcmulr 16669 0gc0g 16816 Σg cgsu 16817 SymGrpcsymg 18613 pmSgncpsgn 18735 mulGrpcmgp 19358 1rcur 19370 CRingccrg 19417 ℤRHomczrh 20320 Mat cmat 21158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-xor 1507 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-ot 4525 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-sup 8979 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-xnn0 12049 df-z 12063 df-dec 12180 df-uz 12325 df-rp 12473 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-hash 13783 df-word 13956 df-lsw 14004 df-concat 14012 df-s1 14039 df-substr 14092 df-pfx 14122 df-splice 14201 df-reverse 14210 df-s2 14299 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-0g 16818 df-gsum 16819 df-prds 16824 df-pws 16826 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-efmnd 18150 df-grp 18222 df-minusg 18223 df-mulg 18343 df-subg 18394 df-ghm 18474 df-gim 18517 df-cntz 18565 df-oppg 18592 df-symg 18614 df-pmtr 18688 df-psgn 18737 df-cmn 19026 df-mgp 19359 df-ur 19371 df-ring 19418 df-cring 19419 df-rnghom 19589 df-subrg 19652 df-sra 20063 df-rgmod 20064 df-cnfld 20218 df-zring 20290 df-zrh 20324 df-dsmm 20548 df-frlm 20563 df-mat 21159 |
This theorem is referenced by: smadiadetlem3 21419 |
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