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Mirrors > Home > MPE Home > Th. List > m2detleiblem6 | Structured version Visualization version GIF version |
Description: Lemma 6 for m2detleib 20437. (Contributed by AV, 20-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) |
m2detleiblem1.i | ⊢ 𝐼 = (invg‘𝑅) |
Ref | Expression |
---|---|
m2detleiblem6 | ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (𝑌‘(𝑆‘𝑄)) = (𝐼‘ 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ex 10035 | . . . . 5 ⊢ 1 ∈ V | |
2 | 2nn 11185 | . . . . 5 ⊢ 2 ∈ ℕ | |
3 | prex 4909 | . . . . . . 7 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ V | |
4 | 3 | prid2 4298 | . . . . . 6 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} |
5 | eqid 2622 | . . . . . . 7 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
6 | m2detleiblem1.p | . . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
7 | m2detleiblem1.n | . . . . . . 7 ⊢ 𝑁 = {1, 2} | |
8 | 5, 6, 7 | symg2bas 17818 | . . . . . 6 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}}) |
9 | 4, 8 | syl5eleqr 2708 | . . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {〈1, 2〉, 〈2, 1〉} ∈ 𝑃) |
10 | 1, 2, 9 | mp2an 708 | . . . 4 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ 𝑃 |
11 | eleq1 2689 | . . . 4 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑄 ∈ 𝑃 ↔ {〈1, 2〉, 〈2, 1〉} ∈ 𝑃)) | |
12 | 10, 11 | mpbiri 248 | . . 3 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → 𝑄 ∈ 𝑃) |
13 | m2detleiblem1.y | . . . 4 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
14 | m2detleiblem1.s | . . . 4 ⊢ 𝑆 = (pmSgn‘𝑁) | |
15 | m2detleiblem1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
16 | 7, 6, 13, 14, 15 | m2detleiblem1 20430 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
17 | 12, 16 | sylan2 491 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
18 | fveq2 6191 | . . . . 5 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{〈1, 2〉, 〈2, 1〉})) | |
19 | 18 | adantl 482 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{〈1, 2〉, 〈2, 1〉})) |
20 | eqid 2622 | . . . . 5 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
21 | eqid 2622 | . . . . 5 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁) | |
22 | 7, 5, 6, 20, 21 | psgnprfval2 17943 | . . . 4 ⊢ ((pmSgn‘𝑁)‘{〈1, 2〉, 〈2, 1〉}) = -1 |
23 | 19, 22 | syl6eq 2672 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → ((pmSgn‘𝑁)‘𝑄) = -1) |
24 | 23 | oveq1d 6665 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 ) = (-1(.g‘𝑅) 1 )) |
25 | ringgrp 18552 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
26 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
27 | 26, 15 | ringidcl 18568 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
28 | eqid 2622 | . . . . 5 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
29 | m2detleiblem1.i | . . . . 5 ⊢ 𝐼 = (invg‘𝑅) | |
30 | 26, 28, 29 | mulgm1 17562 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ (Base‘𝑅)) → (-1(.g‘𝑅) 1 ) = (𝐼‘ 1 )) |
31 | 25, 27, 30 | syl2anc 693 | . . 3 ⊢ (𝑅 ∈ Ring → (-1(.g‘𝑅) 1 ) = (𝐼‘ 1 )) |
32 | 31 | adantr 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (-1(.g‘𝑅) 1 ) = (𝐼‘ 1 )) |
33 | 17, 24, 32 | 3eqtrd 2660 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (𝑌‘(𝑆‘𝑄)) = (𝐼‘ 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {cpr 4179 〈cop 4183 ran crn 5115 ‘cfv 5888 (class class class)co 6650 1c1 9937 -cneg 10267 ℕcn 11020 2c2 11070 Basecbs 15857 Grpcgrp 17422 invgcminusg 17423 .gcmg 17540 SymGrpcsymg 17797 pmTrspcpmtr 17861 pmSgncpsgn 17909 1rcur 18501 Ringcrg 18547 ℤRHomczrh 19848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-oppg 17776 df-symg 17798 df-pmtr 17862 df-psgn 17911 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-rnghom 18715 df-subrg 18778 df-cnfld 19747 df-zring 19819 df-zrh 19852 |
This theorem is referenced by: m2detleib 20437 |
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