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Mirrors > Home > MPE Home > Th. List > zrhpsgnelbas | Structured version Visualization version GIF version |
Description: Embedding of permutation signs into a ring results in an element of the ring. (Contributed by AV, 1-Jan-2019.) |
Ref | Expression |
---|---|
zrhpsgnelbas.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
zrhpsgnelbas.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
zrhpsgnelbas.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
Ref | Expression |
---|---|
zrhpsgnelbas | ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrhpsgnelbas.p | . . . 4 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
2 | zrhpsgnelbas.s | . . . 4 ⊢ 𝑆 = (pmSgn‘𝑁) | |
3 | 1, 2 | psgnran 18626 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1}) |
4 | 3 | 3adant1 1126 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1}) |
5 | elpri 4575 | . . 3 ⊢ ((𝑆‘𝑄) ∈ {1, -1} → ((𝑆‘𝑄) = 1 ∨ (𝑆‘𝑄) = -1)) | |
6 | zrhpsgnelbas.y | . . . . . . . 8 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
7 | eqid 2821 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 6, 7 | zrh1 20643 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑌‘1) = (1r‘𝑅)) |
9 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | 9, 7 | ringidcl 19301 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
11 | 8, 10 | eqeltrd 2913 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑌‘1) ∈ (Base‘𝑅)) |
12 | 11 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘1) ∈ (Base‘𝑅)) |
13 | fveq2 6656 | . . . . . 6 ⊢ ((𝑆‘𝑄) = 1 → (𝑌‘(𝑆‘𝑄)) = (𝑌‘1)) | |
14 | 13 | eleq1d 2897 | . . . . 5 ⊢ ((𝑆‘𝑄) = 1 → ((𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅) ↔ (𝑌‘1) ∈ (Base‘𝑅))) |
15 | 12, 14 | syl5ibr 248 | . . . 4 ⊢ ((𝑆‘𝑄) = 1 → ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅))) |
16 | neg1z 12005 | . . . . . . . 8 ⊢ -1 ∈ ℤ | |
17 | eqid 2821 | . . . . . . . . 9 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
18 | 6, 17, 7 | zrhmulg 20640 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ -1 ∈ ℤ) → (𝑌‘-1) = (-1(.g‘𝑅)(1r‘𝑅))) |
19 | 16, 18 | mpan2 689 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑌‘-1) = (-1(.g‘𝑅)(1r‘𝑅))) |
20 | ringgrp 19285 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
21 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → -1 ∈ ℤ) |
22 | 9, 17, 20, 21, 10 | mulgcld 18232 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (-1(.g‘𝑅)(1r‘𝑅)) ∈ (Base‘𝑅)) |
23 | 19, 22 | eqeltrd 2913 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑌‘-1) ∈ (Base‘𝑅)) |
24 | 23 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘-1) ∈ (Base‘𝑅)) |
25 | fveq2 6656 | . . . . . 6 ⊢ ((𝑆‘𝑄) = -1 → (𝑌‘(𝑆‘𝑄)) = (𝑌‘-1)) | |
26 | 25 | eleq1d 2897 | . . . . 5 ⊢ ((𝑆‘𝑄) = -1 → ((𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅) ↔ (𝑌‘-1) ∈ (Base‘𝑅))) |
27 | 24, 26 | syl5ibr 248 | . . . 4 ⊢ ((𝑆‘𝑄) = -1 → ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅))) |
28 | 15, 27 | jaoi 853 | . . 3 ⊢ (((𝑆‘𝑄) = 1 ∨ (𝑆‘𝑄) = -1) → ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅))) |
29 | 5, 28 | syl 17 | . 2 ⊢ ((𝑆‘𝑄) ∈ {1, -1} → ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅))) |
30 | 4, 29 | mpcom 38 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {cpr 4555 ‘cfv 6341 (class class class)co 7142 Fincfn 8495 1c1 10524 -cneg 10857 ℤcz 11968 Basecbs 16466 .gcmg 18207 SymGrpcsymg 18478 pmSgncpsgn 18600 1rcur 19234 Ringcrg 19280 ℤRHomczrh 20630 |
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 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-addf 10602 ax-mulf 10603 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-ot 4562 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-tpos 7878 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-xnn0 11955 df-z 11969 df-dec 12086 df-uz 12231 df-rp 12377 df-fz 12883 df-fzo 13024 df-seq 13360 df-exp 13420 df-hash 13681 df-word 13852 df-lsw 13900 df-concat 13908 df-s1 13935 df-substr 13988 df-pfx 14018 df-splice 14097 df-reverse 14106 df-s2 14195 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-starv 16563 df-tset 16567 df-ple 16568 df-ds 16570 df-unif 16571 df-0g 16698 df-gsum 16699 df-mre 16840 df-mrc 16841 df-acs 16843 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-mhm 17939 df-submnd 17940 df-efmnd 18017 df-grp 18089 df-minusg 18090 df-mulg 18208 df-subg 18259 df-ghm 18339 df-gim 18382 df-oppg 18457 df-symg 18479 df-pmtr 18553 df-psgn 18602 df-cmn 18891 df-mgp 19223 df-ur 19235 df-ring 19282 df-cring 19283 df-rnghom 19450 df-subrg 19516 df-cnfld 20529 df-zring 20601 df-zrh 20634 |
This theorem is referenced by: zrhcopsgnelbas 20722 m2detleib 21223 mdetpmtr1 31098 |
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