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Mirrors > Home > MPE Home > Th. List > m2detleiblem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for m2detleib 22619. (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 4647 | . . . . 5 ⊢ (𝑄 ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} → (𝑄 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑄 = {〈1, 2〉, 〈2, 1〉})) | |
2 | fveq2 6891 | . . . . . . . 8 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑆‘𝑄) = (𝑆‘{〈1, 1〉, 〈2, 2〉})) | |
3 | m2detleiblem1.n | . . . . . . . . 9 ⊢ 𝑁 = {1, 2} | |
4 | eqid 2726 | . . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
5 | m2detleiblem1.p | . . . . . . . . 9 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
6 | eqid 2726 | . . . . . . . . 9 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
7 | m2detleiblem1.s | . . . . . . . . 9 ⊢ 𝑆 = (pmSgn‘𝑁) | |
8 | 3, 4, 5, 6, 7 | psgnprfval1 19514 | . . . . . . . 8 ⊢ (𝑆‘{〈1, 1〉, 〈2, 2〉}) = 1 |
9 | 2, 8 | eqtrdi 2782 | . . . . . . 7 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑆‘𝑄) = 1) |
10 | 1z 12636 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
11 | 9, 10 | eqeltrdi 2834 | . . . . . 6 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑆‘𝑄) ∈ ℤ) |
12 | fveq2 6891 | . . . . . . . 8 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑆‘𝑄) = (𝑆‘{〈1, 2〉, 〈2, 1〉})) | |
13 | 3, 4, 5, 6, 7 | psgnprfval2 19515 | . . . . . . . 8 ⊢ (𝑆‘{〈1, 2〉, 〈2, 1〉}) = -1 |
14 | 12, 13 | eqtrdi 2782 | . . . . . . 7 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑆‘𝑄) = -1) |
15 | neg1z 12642 | . . . . . . 7 ⊢ -1 ∈ ℤ | |
16 | 14, 15 | eqeltrdi 2834 | . . . . . 6 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑆‘𝑄) ∈ ℤ) |
17 | 11, 16 | jaoi 855 | . . . . 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 11249 | . . . . 5 ⊢ 1 ∈ V | |
20 | 2nn 12329 | . . . . 5 ⊢ 2 ∈ ℕ | |
21 | 4, 5, 3 | symg2bas 19384 | . . . . 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 2844 | . . 3 ⊢ (𝑄 ∈ 𝑃 → (𝑆‘𝑄) ∈ ℤ) |
24 | m2detleiblem1.y | . . . 4 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
25 | eqid 2726 | . . . 4 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
26 | m2detleiblem1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
27 | 24, 25, 26 | zrhmulg 21493 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑆‘𝑄) ∈ ℤ) → (𝑌‘(𝑆‘𝑄)) = ((𝑆‘𝑄)(.g‘𝑅) 1 )) |
28 | 23, 27 | sylan2 591 | . 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 6893 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) = ((pmSgn‘𝑁)‘𝑄)) |
31 | 30 | oveq1d 7429 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → ((𝑆‘𝑄)(.g‘𝑅) 1 ) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
32 | 28, 31 | eqtrd 2766 | 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 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 Vcvv 3463 {cpr 4626 〈cop 4630 ran crn 5674 ‘cfv 6544 (class class class)co 7414 1c1 11148 -cneg 11484 ℕcn 12256 2c2 12311 ℤcz 12602 Basecbs 17206 .gcmg 19055 SymGrpcsymg 19358 pmTrspcpmtr 19433 pmSgncpsgn 19481 1rcur 20158 Ringcrg 20210 ℤRHomczrh 21483 |
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 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-addf 11226 ax-mulf 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 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 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 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 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-oadd 8490 df-er 8724 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9935 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-xnn0 12589 df-z 12603 df-dec 12722 df-uz 12867 df-rp 13021 df-fz 13531 df-fzo 13674 df-seq 14014 df-exp 14074 df-fac 14284 df-bc 14313 df-hash 14341 df-word 14516 df-lsw 14564 df-concat 14572 df-s1 14597 df-substr 14642 df-pfx 14672 df-splice 14751 df-reverse 14760 df-s2 14850 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-starv 17274 df-tset 17278 df-ple 17279 df-ds 17281 df-unif 17282 df-0g 17449 df-gsum 17450 df-mre 17592 df-mrc 17593 df-acs 17595 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-mhm 18766 df-submnd 18767 df-efmnd 18852 df-grp 18924 df-minusg 18925 df-mulg 19056 df-subg 19111 df-ghm 19201 df-gim 19247 df-oppg 19334 df-symg 19359 df-pmtr 19434 df-psgn 19483 df-cmn 19774 df-abl 19775 df-mgp 20112 df-rng 20130 df-ur 20159 df-ring 20212 df-cring 20213 df-rhm 20448 df-subrng 20522 df-subrg 20547 df-cnfld 21338 df-zring 21431 df-zrh 21487 |
This theorem is referenced by: m2detleiblem5 22613 m2detleiblem6 22614 |
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