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Mirrors > Home > MPE Home > Th. List > pmtrdifwrdel | Structured version Visualization version GIF version |
Description: A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set. (Contributed by AV, 15-Jan-2019.) |
Ref | Expression |
---|---|
pmtrdifel.t | ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) |
pmtrdifel.r | ⊢ 𝑅 = ran (pmTrsp‘𝑁) |
Ref | Expression |
---|---|
pmtrdifwrdel | ⊢ ∀𝑤 ∈ Word 𝑇∃𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrdifel.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) | |
2 | pmtrdifel.r | . . . 4 ⊢ 𝑅 = ran (pmTrsp‘𝑁) | |
3 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑗 = 𝑛 → (𝑤‘𝑗) = (𝑤‘𝑛)) | |
4 | 3 | difeq1d 3727 | . . . . . . 7 ⊢ (𝑗 = 𝑛 → ((𝑤‘𝑗) ∖ I ) = ((𝑤‘𝑛) ∖ I )) |
5 | 4 | dmeqd 5326 | . . . . . 6 ⊢ (𝑗 = 𝑛 → dom ((𝑤‘𝑗) ∖ I ) = dom ((𝑤‘𝑛) ∖ I )) |
6 | 5 | fveq2d 6195 | . . . . 5 ⊢ (𝑗 = 𝑛 → ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )) = ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑛) ∖ I ))) |
7 | 6 | cbvmptv 4750 | . . . 4 ⊢ (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) = (𝑛 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑛) ∖ I ))) |
8 | 1, 2, 7 | pmtrdifwrdellem1 17901 | . . 3 ⊢ (𝑤 ∈ Word 𝑇 → (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) ∈ Word 𝑅) |
9 | 1, 2, 7 | pmtrdifwrdellem2 17902 | . . 3 ⊢ (𝑤 ∈ Word 𝑇 → (#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))))) |
10 | 1, 2, 7 | pmtrdifwrdellem3 17903 | . . 3 ⊢ (𝑤 ∈ Word 𝑇 → ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥)) |
11 | fveq2 6191 | . . . . . 6 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (#‘𝑢) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))))) | |
12 | 11 | eqeq2d 2632 | . . . . 5 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → ((#‘𝑤) = (#‘𝑢) ↔ (#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))))) |
13 | fveq1 6190 | . . . . . . . 8 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (𝑢‘𝑖) = ((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)) | |
14 | 13 | fveq1d 6193 | . . . . . . 7 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → ((𝑢‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥)) |
15 | 14 | eqeq2d 2632 | . . . . . 6 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥) ↔ ((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥))) |
16 | 15 | 2ralbidv 2989 | . . . . 5 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥) ↔ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥))) |
17 | 12, 16 | anbi12d 747 | . . . 4 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥)) ↔ ((#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥)))) |
18 | 17 | rspcev 3309 | . . 3 ⊢ (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) ∈ Word 𝑅 ∧ ((#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥))) → ∃𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))) |
19 | 8, 9, 10, 18 | syl12anc 1324 | . 2 ⊢ (𝑤 ∈ Word 𝑇 → ∃𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))) |
20 | 19 | rgen 2922 | 1 ⊢ ∀𝑤 ∈ Word 𝑇∃𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ∖ cdif 3571 {csn 4177 ↦ cmpt 4729 I cid 5023 dom cdm 5114 ran crn 5115 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ..^cfzo 12465 #chash 13117 Word cword 13291 pmTrspcpmtr 17861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-pmtr 17862 |
This theorem is referenced by: (None) |
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