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Mirrors > Home > MPE Home > Th. List > m2detleiblem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for m2detleib 21996. (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 11156 | . . . . 5 ⊢ 1 ∈ V | |
2 | 2nn 12231 | . . . . 5 ⊢ 2 ∈ ℕ | |
3 | prex 5390 | . . . . . . 7 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ V | |
4 | 3 | prid1 4724 | . . . . . 6 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}} |
5 | eqid 2733 | . . . . . . 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 19179 | . . . . . 6 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{⟨1, 1⟩, ⟨2, 2⟩}, {⟨1, 2⟩, ⟨2, 1⟩}}) |
9 | 4, 8 | eleqtrrid 2841 | . . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃) |
10 | 1, 2, 9 | mp2an 691 | . . . 4 ⊢ {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃 |
11 | eleq1 2822 | . . . 4 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → (𝑄 ∈ 𝑃 ↔ {⟨1, 1⟩, ⟨2, 2⟩} ∈ 𝑃)) | |
12 | 10, 11 | mpbiri 258 | . . 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 21989 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
17 | 12, 16 | sylan2 594 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
18 | fveq2 6843 | . . . . 5 ⊢ (𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) | |
19 | 18 | adantl 483 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩})) |
20 | eqid 2733 | . . . . 5 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
21 | eqid 2733 | . . . . 5 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁) | |
22 | 7, 5, 6, 20, 21 | psgnprfval1 19309 | . . . 4 ⊢ ((pmSgn‘𝑁)‘{⟨1, 1⟩, ⟨2, 2⟩}) = 1 |
23 | 19, 22 | eqtrdi 2789 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → ((pmSgn‘𝑁)‘𝑄) = 1) |
24 | 23 | oveq1d 7373 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 ) = (1(.g‘𝑅) 1 )) |
25 | eqid 2733 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
26 | 25, 15 | ringidcl 19994 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
27 | 26 | adantr 482 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → 1 ∈ (Base‘𝑅)) |
28 | eqid 2733 | . . . 4 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
29 | 25, 28 | mulg1 18888 | . . 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 2777 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → (𝑌‘(𝑆‘𝑄)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 {cpr 4589 ⟨cop 4593 ran crn 5635 ‘cfv 6497 (class class class)co 7358 1c1 11057 ℕcn 12158 2c2 12213 Basecbs 17088 .gcmg 18877 SymGrpcsymg 19153 pmTrspcpmtr 19228 pmSgncpsgn 19276 1rcur 19918 Ringcrg 19969 ℤRHomczrh 20916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9842 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-xnn0 12491 df-z 12505 df-dec 12624 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-fac 14180 df-bc 14209 df-hash 14237 df-word 14409 df-lsw 14457 df-concat 14465 df-s1 14490 df-substr 14535 df-pfx 14565 df-splice 14644 df-reverse 14653 df-s2 14743 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-0g 17328 df-gsum 17329 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-efmnd 18684 df-grp 18756 df-minusg 18757 df-mulg 18878 df-subg 18930 df-ghm 19011 df-gim 19054 df-oppg 19129 df-symg 19154 df-pmtr 19229 df-psgn 19278 df-cmn 19569 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-rnghom 20153 df-subrg 20234 df-cnfld 20813 df-zring 20886 df-zrh 20920 |
This theorem is referenced by: m2detleib 21996 |
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