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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 6846 | . . . . . . . 8 ⊢ (𝑗 = 𝑛 → (𝑤‘𝑗) = (𝑤‘𝑛)) | |
4 | 3 | difeq1d 4085 | . . . . . . 7 ⊢ (𝑗 = 𝑛 → ((𝑤‘𝑗) ∖ I ) = ((𝑤‘𝑛) ∖ I )) |
5 | 4 | dmeqd 5865 | . . . . . 6 ⊢ (𝑗 = 𝑛 → dom ((𝑤‘𝑗) ∖ I ) = dom ((𝑤‘𝑛) ∖ I )) |
6 | 5 | fveq2d 6850 | . . . . 5 ⊢ (𝑗 = 𝑛 → ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )) = ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑛) ∖ I ))) |
7 | 6 | cbvmptv 5222 | . . . 4 ⊢ (𝑗 ∈ (0..^(♯‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) = (𝑛 ∈ (0..^(♯‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑛) ∖ I ))) |
8 | 1, 2, 7 | pmtrdifwrdellem1 19271 | . . 3 ⊢ (𝑤 ∈ Word 𝑇 → (𝑗 ∈ (0..^(♯‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) ∈ Word 𝑅) |
9 | 1, 2, 7 | pmtrdifwrdellem2 19272 | . . 3 ⊢ (𝑤 ∈ Word 𝑇 → (♯‘𝑤) = (♯‘(𝑗 ∈ (0..^(♯‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))))) |
10 | 1, 2, 7 | pmtrdifwrdellem3 19273 | . . 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 6846 | . . . . . 6 ⊢ (𝑢 = (𝑗 ∈ (0..^(♯‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (♯‘𝑢) = (♯‘(𝑗 ∈ (0..^(♯‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))))) | |
12 | 11 | eqeq2d 2744 | . . . . 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 6845 | . . . . . . . 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 6848 | . . . . . . 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 2744 | . . . . . 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 3209 | . . . . 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 632 | . . . 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 3583 | . . 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 836 | . 2 ⊢ (𝑤 ∈ Word 𝑇 → ∃𝑢 ∈ Word 𝑅((♯‘𝑤) = (♯‘𝑢) ∧ ∀𝑖 ∈ (0..^(♯‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))) |
20 | 19 | rgen 3063 | 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 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 ∖ cdif 3911 {csn 4590 ↦ cmpt 5192 I cid 5534 dom cdm 5637 ran crn 5638 ‘cfv 6500 (class class class)co 7361 0cc0 11059 ..^cfzo 13576 ♯chash 14239 Word cword 14411 pmTrspcpmtr 19231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-pmtr 19232 |
This theorem is referenced by: (None) |
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