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Mirrors > Home > MPE Home > Th. List > zrhcopsgnelbas | 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.) (Proof shortened by AV, 3-Jul-2022.) |
Ref | Expression |
---|---|
zrhpsgnelbas.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
zrhpsgnelbas.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
zrhpsgnelbas.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
Ref | Expression |
---|---|
zrhcopsgnelbas | ⊢ ((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 | cofipsgn 21585 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) = (𝑌‘(𝑆‘𝑄))) |
4 | 3 | 3adant1 1127 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) = (𝑌‘(𝑆‘𝑄))) |
5 | zrhpsgnelbas.y | . . 3 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
6 | 1, 2, 5 | zrhpsgnelbas 21586 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) ∈ (Base‘𝑅)) |
7 | 4, 6 | eqeltrd 2826 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∘ ccom 5678 ‘cfv 6546 Fincfn 8966 Basecbs 17208 SymGrpcsymg 19360 pmSgncpsgn 19483 Ringcrg 20212 ℤRHomczrh 21485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-addf 11228 ax-mulf 11229 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-xnn0 12591 df-z 12605 df-dec 12724 df-uz 12869 df-rp 13023 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-word 14518 df-lsw 14566 df-concat 14574 df-s1 14599 df-substr 14644 df-pfx 14674 df-splice 14753 df-reverse 14762 df-s2 14852 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-0g 17451 df-gsum 17452 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-submnd 18769 df-efmnd 18854 df-grp 18926 df-minusg 18927 df-mulg 19058 df-subg 19113 df-ghm 19203 df-gim 19249 df-oppg 19336 df-symg 19361 df-pmtr 19436 df-psgn 19485 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-rhm 20450 df-subrng 20524 df-subrg 20549 df-cnfld 21340 df-zring 21433 df-zrh 21489 |
This theorem is referenced by: madetsmelbas 22454 madetsmelbas2 22455 mdet0pr 22582 smadiadetlem1a 22653 mdetpmtr12 33653 |
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