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Mirrors > Home > MPE Home > Th. List > m2detleiblem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for m2detleib 21234. (Contributed by AV, 12-Dec-2018.) |
Ref | Expression |
---|---|
m2detleiblem1.n | ⊢ 𝑁 = {1, 2} |
m2detleiblem1.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
m2detleiblem1.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
m2detleiblem1.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
m2detleiblem1.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
m2detleiblem1 | ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 4582 | . . . . 5 ⊢ (𝑄 ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} → (𝑄 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑄 = {〈1, 2〉, 〈2, 1〉})) | |
2 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑆‘𝑄) = (𝑆‘{〈1, 1〉, 〈2, 2〉})) | |
3 | m2detleiblem1.n | . . . . . . . . 9 ⊢ 𝑁 = {1, 2} | |
4 | eqid 2821 | . . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
5 | m2detleiblem1.p | . . . . . . . . 9 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
6 | eqid 2821 | . . . . . . . . 9 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
7 | m2detleiblem1.s | . . . . . . . . 9 ⊢ 𝑆 = (pmSgn‘𝑁) | |
8 | 3, 4, 5, 6, 7 | psgnprfval1 18644 | . . . . . . . 8 ⊢ (𝑆‘{〈1, 1〉, 〈2, 2〉}) = 1 |
9 | 2, 8 | syl6eq 2872 | . . . . . . 7 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑆‘𝑄) = 1) |
10 | 1z 12006 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
11 | 9, 10 | eqeltrdi 2921 | . . . . . 6 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑆‘𝑄) ∈ ℤ) |
12 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑆‘𝑄) = (𝑆‘{〈1, 2〉, 〈2, 1〉})) | |
13 | 3, 4, 5, 6, 7 | psgnprfval2 18645 | . . . . . . . 8 ⊢ (𝑆‘{〈1, 2〉, 〈2, 1〉}) = -1 |
14 | 12, 13 | syl6eq 2872 | . . . . . . 7 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑆‘𝑄) = -1) |
15 | neg1z 12012 | . . . . . . 7 ⊢ -1 ∈ ℤ | |
16 | 14, 15 | eqeltrdi 2921 | . . . . . 6 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑆‘𝑄) ∈ ℤ) |
17 | 11, 16 | jaoi 853 | . . . . 5 ⊢ ((𝑄 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (𝑆‘𝑄) ∈ ℤ) |
18 | 1, 17 | syl 17 | . . . 4 ⊢ (𝑄 ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} → (𝑆‘𝑄) ∈ ℤ) |
19 | 1ex 10631 | . . . . 5 ⊢ 1 ∈ V | |
20 | 2nn 11704 | . . . . 5 ⊢ 2 ∈ ℕ | |
21 | 4, 5, 3 | symg2bas 18515 | . . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}}) |
22 | 19, 20, 21 | mp2an 690 | . . . 4 ⊢ 𝑃 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} |
23 | 18, 22 | eleq2s 2931 | . . 3 ⊢ (𝑄 ∈ 𝑃 → (𝑆‘𝑄) ∈ ℤ) |
24 | m2detleiblem1.y | . . . 4 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
25 | eqid 2821 | . . . 4 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
26 | m2detleiblem1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
27 | 24, 25, 26 | zrhmulg 20651 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑆‘𝑄) ∈ ℤ) → (𝑌‘(𝑆‘𝑄)) = ((𝑆‘𝑄)(.g‘𝑅) 1 )) |
28 | 23, 27 | sylan2 594 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = ((𝑆‘𝑄)(.g‘𝑅) 1 )) |
29 | 7 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → 𝑆 = (pmSgn‘𝑁)) |
30 | 29 | fveq1d 6666 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) = ((pmSgn‘𝑁)‘𝑄)) |
31 | 30 | oveq1d 7165 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → ((𝑆‘𝑄)(.g‘𝑅) 1 ) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
32 | 28, 31 | eqtrd 2856 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {cpr 4562 〈cop 4566 ran crn 5550 ‘cfv 6349 (class class class)co 7150 1c1 10532 -cneg 10865 ℕcn 11632 2c2 11686 ℤcz 11975 Basecbs 16477 .gcmg 18218 SymGrpcsymg 18489 pmTrspcpmtr 18563 pmSgncpsgn 18611 1rcur 19245 Ringcrg 19291 ℤRHomczrh 20641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1501 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-splice 14106 df-reverse 14115 df-s2 14204 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-efmnd 18028 df-grp 18100 df-minusg 18101 df-mulg 18219 df-subg 18270 df-ghm 18350 df-gim 18393 df-oppg 18468 df-symg 18490 df-pmtr 18564 df-psgn 18613 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-rnghom 19461 df-subrg 19527 df-cnfld 20540 df-zring 20612 df-zrh 20645 |
This theorem is referenced by: m2detleiblem5 21228 m2detleiblem6 21229 |
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