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Mirrors > Home > MPE Home > Th. List > m2detleiblem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for m2detleib 21778. (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) |
Ref | Expression |
---|---|
m2detleiblem5 | ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → (𝑌‘(𝑆‘𝑄)) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ex 10972 | . . . . 5 ⊢ 1 ∈ V | |
2 | 2nn 12046 | . . . . 5 ⊢ 2 ∈ ℕ | |
3 | prex 5359 | . . . . . . 7 ⊢ {〈1, 1〉, 〈2, 2〉} ∈ V | |
4 | 3 | prid1 4704 | . . . . . 6 ⊢ {〈1, 1〉, 〈2, 2〉} ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} |
5 | eqid 2740 | . . . . . . 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 18998 | . . . . . 6 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}}) |
9 | 4, 8 | eleqtrrid 2848 | . . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {〈1, 1〉, 〈2, 2〉} ∈ 𝑃) |
10 | 1, 2, 9 | mp2an 689 | . . . 4 ⊢ {〈1, 1〉, 〈2, 2〉} ∈ 𝑃 |
11 | eleq1 2828 | . . . 4 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑄 ∈ 𝑃 ↔ {〈1, 1〉, 〈2, 2〉} ∈ 𝑃)) | |
12 | 10, 11 | mpbiri 257 | . . 3 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → 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 21771 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
17 | 12, 16 | sylan2 593 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
18 | fveq2 6771 | . . . . 5 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{〈1, 1〉, 〈2, 2〉})) | |
19 | 18 | adantl 482 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{〈1, 1〉, 〈2, 2〉})) |
20 | eqid 2740 | . . . . 5 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
21 | eqid 2740 | . . . . 5 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁) | |
22 | 7, 5, 6, 20, 21 | psgnprfval1 19128 | . . . 4 ⊢ ((pmSgn‘𝑁)‘{〈1, 1〉, 〈2, 2〉}) = 1 |
23 | 19, 22 | eqtrdi 2796 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → ((pmSgn‘𝑁)‘𝑄) = 1) |
24 | 23 | oveq1d 7286 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 ) = (1(.g‘𝑅) 1 )) |
25 | eqid 2740 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
26 | 25, 15 | ringidcl 19805 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
27 | 26 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → 1 ∈ (Base‘𝑅)) |
28 | eqid 2740 | . . . 4 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
29 | 25, 28 | mulg1 18709 | . . 3 ⊢ ( 1 ∈ (Base‘𝑅) → (1(.g‘𝑅) 1 ) = 1 ) |
30 | 27, 29 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → (1(.g‘𝑅) 1 ) = 1 ) |
31 | 17, 24, 30 | 3eqtrd 2784 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → (𝑌‘(𝑆‘𝑄)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 {cpr 4569 〈cop 4573 ran crn 5591 ‘cfv 6432 (class class class)co 7271 1c1 10873 ℕcn 11973 2c2 12028 Basecbs 16910 .gcmg 18698 SymGrpcsymg 18972 pmTrspcpmtr 19047 pmSgncpsgn 19095 1rcur 19735 Ringcrg 19781 ℤRHomczrh 20699 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-oadd 8292 df-er 8481 df-map 8600 df-pm 8601 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-dju 9660 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12437 df-uz 12582 df-rp 12730 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-fac 13986 df-bc 14015 df-hash 14043 df-word 14216 df-lsw 14264 df-concat 14272 df-s1 14299 df-substr 14352 df-pfx 14382 df-splice 14461 df-reverse 14470 df-s2 14559 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-0g 17150 df-gsum 17151 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-efmnd 18506 df-grp 18578 df-minusg 18579 df-mulg 18699 df-subg 18750 df-ghm 18830 df-gim 18873 df-oppg 18948 df-symg 18973 df-pmtr 19048 df-psgn 19097 df-cmn 19386 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-rnghom 19957 df-subrg 20020 df-cnfld 20596 df-zring 20669 df-zrh 20703 |
This theorem is referenced by: m2detleib 21778 |
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