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Mirrors > Home > MPE Home > Th. List > zrhcofipsgn | Structured version Visualization version GIF version |
Description: Composition of a ℤRHom homomorphism and the sign function for a finite permutation. (Contributed by AV, 27-Dec-2018.) |
Ref | Expression |
---|---|
zrhcofipsgn.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
zrhcofipsgn.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
zrhcofipsgn.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
Ref | Expression |
---|---|
zrhcofipsgn | ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) = (𝑌‘(𝑆‘𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | zrhcofipsgn.p | . . 3 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
3 | eqid 2610 | . . 3 ⊢ {𝑝 ∈ 𝑃 ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ 𝑃 ∣ dom (𝑝 ∖ I ) ∈ Fin} | |
4 | zrhcofipsgn.s | . . 3 ⊢ 𝑆 = (pmSgn‘𝑁) | |
5 | 1, 2, 3, 4 | psgnfn 17744 | . 2 ⊢ 𝑆 Fn {𝑝 ∈ 𝑃 ∣ dom (𝑝 ∖ I ) ∈ Fin} |
6 | simpr 476 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → 𝑄 ∈ 𝑃) | |
7 | 1, 2 | sygbasnfpfi 17755 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → dom (𝑄 ∖ I ) ∈ Fin) |
8 | difeq1 3683 | . . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝 ∖ I ) = (𝑄 ∖ I )) | |
9 | 8 | dmeqd 5248 | . . . . 5 ⊢ (𝑝 = 𝑄 → dom (𝑝 ∖ I ) = dom (𝑄 ∖ I )) |
10 | 9 | eleq1d 2672 | . . . 4 ⊢ (𝑝 = 𝑄 → (dom (𝑝 ∖ I ) ∈ Fin ↔ dom (𝑄 ∖ I ) ∈ Fin)) |
11 | 10 | elrab 3331 | . . 3 ⊢ (𝑄 ∈ {𝑝 ∈ 𝑃 ∣ dom (𝑝 ∖ I ) ∈ Fin} ↔ (𝑄 ∈ 𝑃 ∧ dom (𝑄 ∖ I ) ∈ Fin)) |
12 | 6, 7, 11 | sylanbrc 695 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → 𝑄 ∈ {𝑝 ∈ 𝑃 ∣ dom (𝑝 ∖ I ) ∈ Fin}) |
13 | fvco2 6183 | . 2 ⊢ ((𝑆 Fn {𝑝 ∈ 𝑃 ∣ dom (𝑝 ∖ I ) ∈ Fin} ∧ 𝑄 ∈ {𝑝 ∈ 𝑃 ∣ dom (𝑝 ∖ I ) ∈ Fin}) → ((𝑌 ∘ 𝑆)‘𝑄) = (𝑌‘(𝑆‘𝑄))) | |
14 | 5, 12, 13 | sylancr 694 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) = (𝑌‘(𝑆‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 I cid 4948 dom cdm 5038 ∘ ccom 5042 Fn wfn 5799 ‘cfv 5804 Fincfn 7841 Basecbs 15695 SymGrpcsymg 17620 pmSgncpsgn 17732 ℤRHomczrh 19667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-tset 15787 df-symg 17621 df-psgn 17734 |
This theorem is referenced by: zrhcopsgnelbas 19760 zrhcopsgndif 19768 mdetfval1 20215 mdetpmtr1 29217 mdetpmtr12 29219 |
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