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Theorem pmtrdifwrdel 17728
 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.)
Hypotheses
Ref Expression
pmtrdifel.t 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
pmtrdifel.r 𝑅 = ran (pmTrsp‘𝑁)
Assertion
Ref Expression
pmtrdifwrdel 𝑤 ∈ Word 𝑇𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥))
Distinct variable groups:   𝑥,𝑁   𝑥,𝑇   𝑢,𝐾   𝑖,𝑁,𝑢   𝑇,𝑖   𝑅,𝑖,𝑢   𝑤,𝑖,𝑥,𝑢
Allowed substitution hints:   𝑅(𝑥,𝑤)   𝑇(𝑤,𝑢)   𝐾(𝑥,𝑤,𝑖)   𝑁(𝑤)

Proof of Theorem pmtrdifwrdel
Dummy variables 𝑗 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pmtrdifel.t . . . 4 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
2 pmtrdifel.r . . . 4 𝑅 = ran (pmTrsp‘𝑁)
3 fveq2 6103 . . . . . . . 8 (𝑗 = 𝑛 → (𝑤𝑗) = (𝑤𝑛))
43difeq1d 3689 . . . . . . 7 (𝑗 = 𝑛 → ((𝑤𝑗) ∖ I ) = ((𝑤𝑛) ∖ I ))
54dmeqd 5248 . . . . . 6 (𝑗 = 𝑛 → dom ((𝑤𝑗) ∖ I ) = dom ((𝑤𝑛) ∖ I ))
65fveq2d 6107 . . . . 5 (𝑗 = 𝑛 → ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )) = ((pmTrsp‘𝑁)‘dom ((𝑤𝑛) ∖ I )))
76cbvmptv 4678 . . . 4 (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) = (𝑛 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑛) ∖ I )))
81, 2, 7pmtrdifwrdellem1 17724 . . 3 (𝑤 ∈ Word 𝑇 → (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) ∈ Word 𝑅)
91, 2, 7pmtrdifwrdellem2 17725 . . 3 (𝑤 ∈ Word 𝑇 → (#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))))
101, 2, 7pmtrdifwrdellem3 17726 . . 3 (𝑤 ∈ Word 𝑇 → ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))‘𝑖)‘𝑥))
11 fveq2 6103 . . . . . 6 (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) → (#‘𝑢) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))))
1211eqeq2d 2620 . . . . 5 (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) → ((#‘𝑤) = (#‘𝑢) ↔ (#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))))))
13 fveq1 6102 . . . . . . . 8 (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) → (𝑢𝑖) = ((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))‘𝑖))
1413fveq1d 6105 . . . . . . 7 (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) → ((𝑢𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))‘𝑖)‘𝑥))
1514eqeq2d 2620 . . . . . 6 (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) → (((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥) ↔ ((𝑤𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))‘𝑖)‘𝑥)))
16152ralbidv 2972 . . . . 5 (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) → (∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥) ↔ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))‘𝑖)‘𝑥)))
1712, 16anbi12d 743 . . . 4 (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) → (((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥)) ↔ ((#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))‘𝑖)‘𝑥))))
1817rspcev 3282 . . 3 (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I ))) ∈ Word 𝑅 ∧ ((#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤𝑗) ∖ I )))‘𝑖)‘𝑥))) → ∃𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥)))
198, 9, 10, 18syl12anc 1316 . 2 (𝑤 ∈ Word 𝑇 → ∃𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥)))
2019rgen 2906 1 𝑤 ∈ Word 𝑇𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537  {csn 4125   ↦ cmpt 4643   I cid 4948  dom cdm 5038  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ..^cfzo 12334  #chash 12979  Word cword 13146  pmTrspcpmtr 17684 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-2o 7448  df-oadd 7451  df-er 7629  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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-pmtr 17685 This theorem is referenced by: (None)
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