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Mirrors > Home > MPE Home > Th. List > m2detleiblem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for m2detleib 20805. (Contributed by AV, 12-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 |
---|---|
m2detleiblem1 | ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 4419 | . . . . 5 ⊢ (𝑄 ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} → (𝑄 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑄 = {〈1, 2〉, 〈2, 1〉})) | |
2 | fveq2 6433 | . . . . . . . 8 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑆‘𝑄) = (𝑆‘{〈1, 1〉, 〈2, 2〉})) | |
3 | m2detleiblem1.n | . . . . . . . . 9 ⊢ 𝑁 = {1, 2} | |
4 | eqid 2825 | . . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
5 | m2detleiblem1.p | . . . . . . . . 9 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
6 | eqid 2825 | . . . . . . . . 9 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
7 | m2detleiblem1.s | . . . . . . . . 9 ⊢ 𝑆 = (pmSgn‘𝑁) | |
8 | 3, 4, 5, 6, 7 | psgnprfval1 18293 | . . . . . . . 8 ⊢ (𝑆‘{〈1, 1〉, 〈2, 2〉}) = 1 |
9 | 2, 8 | syl6eq 2877 | . . . . . . 7 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑆‘𝑄) = 1) |
10 | 1z 11735 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
11 | 9, 10 | syl6eqel 2914 | . . . . . 6 ⊢ (𝑄 = {〈1, 1〉, 〈2, 2〉} → (𝑆‘𝑄) ∈ ℤ) |
12 | fveq2 6433 | . . . . . . . 8 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑆‘𝑄) = (𝑆‘{〈1, 2〉, 〈2, 1〉})) | |
13 | 3, 4, 5, 6, 7 | psgnprfval2 18294 | . . . . . . . 8 ⊢ (𝑆‘{〈1, 2〉, 〈2, 1〉}) = -1 |
14 | 12, 13 | syl6eq 2877 | . . . . . . 7 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑆‘𝑄) = -1) |
15 | neg1z 11741 | . . . . . . 7 ⊢ -1 ∈ ℤ | |
16 | 14, 15 | syl6eqel 2914 | . . . . . 6 ⊢ (𝑄 = {〈1, 2〉, 〈2, 1〉} → (𝑆‘𝑄) ∈ ℤ) |
17 | 11, 16 | jaoi 888 | . . . . 5 ⊢ ((𝑄 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑄 = {〈1, 2〉, 〈2, 1〉}) → (𝑆‘𝑄) ∈ ℤ) |
18 | 1, 17 | syl 17 | . . . 4 ⊢ (𝑄 ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} → (𝑆‘𝑄) ∈ ℤ) |
19 | 1ex 10352 | . . . . 5 ⊢ 1 ∈ V | |
20 | 2nn 11424 | . . . . 5 ⊢ 2 ∈ ℕ | |
21 | 4, 5, 3 | symg2bas 18168 | . . . . 5 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝑃 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}}) |
22 | 19, 20, 21 | mp2an 683 | . . . 4 ⊢ 𝑃 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} |
23 | 18, 22 | eleq2s 2924 | . . 3 ⊢ (𝑄 ∈ 𝑃 → (𝑆‘𝑄) ∈ ℤ) |
24 | m2detleiblem1.y | . . . 4 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
25 | eqid 2825 | . . . 4 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
26 | m2detleiblem1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
27 | 24, 25, 26 | zrhmulg 20218 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑆‘𝑄) ∈ ℤ) → (𝑌‘(𝑆‘𝑄)) = ((𝑆‘𝑄)(.g‘𝑅) 1 )) |
28 | 23, 27 | sylan2 586 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = ((𝑆‘𝑄)(.g‘𝑅) 1 )) |
29 | 7 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → 𝑆 = (pmSgn‘𝑁)) |
30 | 29 | fveq1d 6435 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) = ((pmSgn‘𝑁)‘𝑄)) |
31 | 30 | oveq1d 6920 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → ((𝑆‘𝑄)(.g‘𝑅) 1 ) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
32 | 28, 31 | eqtrd 2861 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 878 = wceq 1656 ∈ wcel 2164 Vcvv 3414 {cpr 4399 〈cop 4403 ran crn 5343 ‘cfv 6123 (class class class)co 6905 1c1 10253 -cneg 10586 ℕcn 11350 2c2 11406 ℤcz 11704 Basecbs 16222 .gcmg 17894 SymGrpcsymg 18147 pmTrspcpmtr 18211 pmSgncpsgn 18259 1rcur 18855 Ringcrg 18901 ℤRHomczrh 20208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-xor 1638 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-ot 4406 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-xnn0 11691 df-z 11705 df-dec 11822 df-uz 11969 df-rp 12113 df-fz 12620 df-fzo 12761 df-seq 13096 df-exp 13155 df-fac 13354 df-bc 13383 df-hash 13411 df-word 13575 df-lsw 13623 df-concat 13631 df-s1 13656 df-substr 13701 df-pfx 13750 df-splice 13857 df-reverse 13875 df-s2 13969 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-0g 16455 df-gsum 16456 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-submnd 17689 df-grp 17779 df-minusg 17780 df-mulg 17895 df-subg 17942 df-ghm 18009 df-gim 18052 df-oppg 18126 df-symg 18148 df-pmtr 18212 df-psgn 18261 df-cmn 18548 df-mgp 18844 df-ur 18856 df-ring 18903 df-cring 18904 df-rnghom 19071 df-subrg 19134 df-cnfld 20107 df-zring 20179 df-zrh 20212 |
This theorem is referenced by: m2detleiblem5 20799 m2detleiblem6 20800 |
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