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Mirrors > Home > MPE Home > Th. List > m2detleiblem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for m2detleib 20806. (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 10353 | . . . . 5 ⊢ 1 ∈ V | |
2 | 2nn 11425 | . . . . 5 ⊢ 2 ∈ ℕ | |
3 | prex 5131 | . . . . . . 7 ⊢ {〈1, 1〉, 〈2, 2〉} ∈ V | |
4 | 3 | prid1 4516 | . . . . . 6 ⊢ {〈1, 1〉, 〈2, 2〉} ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} |
5 | eqid 2826 | . . . . . . 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 18169 | . . . . . 6 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}}) |
9 | 4, 8 | syl5eleqr 2914 | . . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → {〈1, 1〉, 〈2, 2〉} ∈ 𝑃) |
10 | 1, 2, 9 | mp2an 685 | . . . 4 ⊢ {〈1, 1〉, 〈2, 2〉} ∈ 𝑃 |
11 | eleq1 2895 | . . . 4 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑄 ∈ 𝑃 ↔ {〈1, 1〉, 〈2, 2〉} ∈ 𝑃)) | |
12 | 10, 11 | mpbiri 250 | . . 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 20799 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
17 | 12, 16 | sylan2 588 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
18 | fveq2 6434 | . . . . 5 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{〈1, 1〉, 〈2, 2〉})) | |
19 | 18 | adantl 475 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → ((pmSgn‘𝑁)‘𝑄) = ((pmSgn‘𝑁)‘{〈1, 1〉, 〈2, 2〉})) |
20 | eqid 2826 | . . . . 5 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
21 | eqid 2826 | . . . . 5 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁) | |
22 | 7, 5, 6, 20, 21 | psgnprfval1 18294 | . . . 4 ⊢ ((pmSgn‘𝑁)‘{〈1, 1〉, 〈2, 2〉}) = 1 |
23 | 19, 22 | syl6eq 2878 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → ((pmSgn‘𝑁)‘𝑄) = 1) |
24 | 23 | oveq1d 6921 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 ) = (1(.g‘𝑅) 1 )) |
25 | eqid 2826 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
26 | 25, 15 | ringidcl 18923 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
27 | 26 | adantr 474 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → 1 ∈ (Base‘𝑅)) |
28 | eqid 2826 | . . . 4 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
29 | 25, 28 | mulg1 17903 | . . 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 2866 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {〈1, 1〉, 〈2, 2〉}) → (𝑌‘(𝑆‘𝑄)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Vcvv 3415 {cpr 4400 〈cop 4404 ran crn 5344 ‘cfv 6124 (class class class)co 6906 1c1 10254 ℕcn 11351 2c2 11407 Basecbs 16223 .gcmg 17895 SymGrpcsymg 18148 pmTrspcpmtr 18212 pmSgncpsgn 18260 1rcur 18856 Ringcrg 18902 ℤRHomczrh 20209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-xor 1640 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-2o 7828 df-oadd 7831 df-er 8010 df-map 8125 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-card 9079 df-cda 9306 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-xnn0 11692 df-z 11706 df-dec 11823 df-uz 11970 df-rp 12114 df-fz 12621 df-fzo 12762 df-seq 13097 df-exp 13156 df-fac 13355 df-bc 13384 df-hash 13412 df-word 13576 df-lsw 13624 df-concat 13632 df-s1 13657 df-substr 13702 df-pfx 13751 df-splice 13858 df-reverse 13876 df-s2 13970 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-0g 16456 df-gsum 16457 df-mre 16600 df-mrc 16601 df-acs 16603 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-mhm 17689 df-submnd 17690 df-grp 17780 df-minusg 17781 df-mulg 17896 df-subg 17943 df-ghm 18010 df-gim 18053 df-oppg 18127 df-symg 18149 df-pmtr 18213 df-psgn 18262 df-cmn 18549 df-mgp 18845 df-ur 18857 df-ring 18904 df-cring 18905 df-rnghom 19072 df-subrg 19135 df-cnfld 20108 df-zring 20180 df-zrh 20213 |
This theorem is referenced by: m2detleib 20806 |
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