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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 19474 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1}) |
4 | 3 | 3adant1 1127 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1}) |
5 | elpri 4647 | . . 3 ⊢ ((𝑆‘𝑄) ∈ {1, -1} → ((𝑆‘𝑄) = 1 ∨ (𝑆‘𝑄) = -1)) | |
6 | zrhpsgnelbas.y | . . . . . . . 8 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
7 | eqid 2725 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 6, 7 | zrh1 21442 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑌‘1) = (1r‘𝑅)) |
9 | eqid 2725 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | 9, 7 | ringidcl 20206 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
11 | 8, 10 | eqeltrd 2825 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑌‘1) ∈ (Base‘𝑅)) |
12 | 11 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘1) ∈ (Base‘𝑅)) |
13 | fveq2 6892 | . . . . . 6 ⊢ ((𝑆‘𝑄) = 1 → (𝑌‘(𝑆‘𝑄)) = (𝑌‘1)) | |
14 | 13 | eleq1d 2810 | . . . . 5 ⊢ ((𝑆‘𝑄) = 1 → ((𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅) ↔ (𝑌‘1) ∈ (Base‘𝑅))) |
15 | 12, 14 | imbitrrid 245 | . . . 4 ⊢ ((𝑆‘𝑄) = 1 → ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅))) |
16 | neg1z 12628 | . . . . . . . 8 ⊢ -1 ∈ ℤ | |
17 | eqid 2725 | . . . . . . . . 9 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
18 | 6, 17, 7 | zrhmulg 21439 | . . . . . . . 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 20182 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
21 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → -1 ∈ ℤ) |
22 | 9, 17, 20, 21, 10 | mulgcld 19055 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (-1(.g‘𝑅)(1r‘𝑅)) ∈ (Base‘𝑅)) |
23 | 19, 22 | eqeltrd 2825 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑌‘-1) ∈ (Base‘𝑅)) |
24 | 23 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘-1) ∈ (Base‘𝑅)) |
25 | fveq2 6892 | . . . . . 6 ⊢ ((𝑆‘𝑄) = -1 → (𝑌‘(𝑆‘𝑄)) = (𝑌‘-1)) | |
26 | 25 | eleq1d 2810 | . . . . 5 ⊢ ((𝑆‘𝑄) = -1 → ((𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅) ↔ (𝑌‘-1) ∈ (Base‘𝑅))) |
27 | 24, 26 | imbitrrid 245 | . . . 4 ⊢ ((𝑆‘𝑄) = -1 → ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅))) |
28 | 15, 27 | jaoi 855 | . . 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 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {cpr 4626 ‘cfv 6543 (class class class)co 7416 Fincfn 8962 1c1 11139 -cneg 11475 ℤcz 12588 Basecbs 17179 .gcmg 19027 SymGrpcsymg 19325 pmSgncpsgn 19448 1rcur 20125 Ringcrg 20177 ℤRHomczrh 21429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-addf 11217 ax-mulf 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-word 14497 df-lsw 14545 df-concat 14553 df-s1 14578 df-substr 14623 df-pfx 14653 df-splice 14732 df-reverse 14741 df-s2 14831 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-0g 17422 df-gsum 17423 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-efmnd 18825 df-grp 18897 df-minusg 18898 df-mulg 19028 df-subg 19082 df-ghm 19172 df-gim 19217 df-oppg 19301 df-symg 19326 df-pmtr 19401 df-psgn 19450 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-cnfld 21284 df-zring 21377 df-zrh 21433 |
This theorem is referenced by: zrhcopsgnelbas 21531 m2detleib 22551 mdetpmtr1 33481 |
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