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Mirrors > Home > MPE Home > Th. List > m2detleiblem6 | Structured version Visualization version GIF version |
Description: Lemma 6 for m2detleib 21242. (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 10639 | . . . . 5 ⊢ 1 ∈ V | |
2 | 2nn 11713 | . . . . 5 ⊢ 2 ∈ ℕ | |
3 | prex 5335 | . . . . . . 7 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ V | |
4 | 3 | prid2 4701 | . . . . . 6 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} |
5 | eqid 2823 | . . . . . . 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 18523 | . . . . . 6 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}}) |
9 | 4, 8 | eleqtrrid 2922 | . . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {〈1, 2〉, 〈2, 1〉} ∈ 𝑃) |
10 | 1, 2, 9 | mp2an 690 | . . . 4 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ 𝑃 |
11 | eleq1 2902 | . . . 4 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑄 ∈ 𝑃 ↔ {〈1, 2〉, 〈2, 1〉} ∈ 𝑃)) | |
12 | 10, 11 | mpbiri 260 | . . 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 21235 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
17 | 12, 16 | sylan2 594 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
18 | fveq2 6672 | . . . . 5 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{〈1, 2〉, 〈2, 1〉})) | |
19 | 18 | adantl 484 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{〈1, 2〉, 〈2, 1〉})) |
20 | eqid 2823 | . . . . 5 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
21 | eqid 2823 | . . . . 5 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁) | |
22 | 7, 5, 6, 20, 21 | psgnprfval2 18653 | . . . 4 ⊢ ((pmSgn‘𝑁)‘{〈1, 2〉, 〈2, 1〉}) = -1 |
23 | 19, 22 | syl6eq 2874 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → ((pmSgn‘𝑁)‘𝑄) = -1) |
24 | 23 | oveq1d 7173 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 ) = (-1(.g‘𝑅) 1 )) |
25 | ringgrp 19304 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
26 | eqid 2823 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
27 | 26, 15 | ringidcl 19320 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
28 | eqid 2823 | . . . . 5 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
29 | m2detleiblem1.i | . . . . 5 ⊢ 𝐼 = (invg‘𝑅) | |
30 | 26, 28, 29 | mulgm1 18250 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ (Base‘𝑅)) → (-1(.g‘𝑅) 1 ) = (𝐼‘ 1 )) |
31 | 25, 27, 30 | syl2anc 586 | . . 3 ⊢ (𝑅 ∈ Ring → (-1(.g‘𝑅) 1 ) = (𝐼‘ 1 )) |
32 | 31 | adantr 483 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (-1(.g‘𝑅) 1 ) = (𝐼‘ 1 )) |
33 | 17, 24, 32 | 3eqtrd 2862 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (𝑌‘(𝑆‘𝑄)) = (𝐼‘ 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {cpr 4571 〈cop 4575 ran crn 5558 ‘cfv 6357 (class class class)co 7158 1c1 10540 -cneg 10873 ℕcn 11640 2c2 11695 Basecbs 16485 Grpcgrp 18105 invgcminusg 18106 .gcmg 18226 SymGrpcsymg 18497 pmTrspcpmtr 18571 pmSgncpsgn 18619 1rcur 19253 Ringcrg 19299 ℤRHomczrh 20649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-word 13865 df-lsw 13917 df-concat 13925 df-s1 13952 df-substr 14005 df-pfx 14035 df-splice 14114 df-reverse 14123 df-s2 14212 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-efmnd 18036 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-oppg 18476 df-symg 18498 df-pmtr 18572 df-psgn 18621 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-rnghom 19469 df-subrg 19535 df-cnfld 20548 df-zring 20620 df-zrh 20653 |
This theorem is referenced by: m2detleib 21242 |
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