Theorem pmtrdifwrdel2 17729

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 not moving the special element. (Contributed by AV, 31-Jan-2019.)

Hypotheses

Ref	Expression
pmtrdifel.t	⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
pmtrdifel.r	⊢ 𝑅 = ran (pmTrsp‘𝑁)

Assertion

Ref	Expression
pmtrdifwrdel2	⊢ (𝐾 ∈ 𝑁 → ∀𝑤 ∈ Word 𝑇∃𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑢‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))))

Distinct variable groups:   𝑥,𝑁   𝑥,𝑇   𝑢,𝐾   𝑖,𝑁,𝑢   𝑇,𝑖   𝑅,𝑖,𝑢   𝑤,𝑖,𝑥,𝑢   𝑖,𝐾,𝑤   𝑤,𝑁

Allowed substitution hints:   𝑅(𝑥,𝑤)   𝑇(𝑤,𝑢)   𝐾(𝑥)

Proof of Theorem pmtrdifwrdel2

Dummy variables 𝑗 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmtrdifel.t	. . . . 5 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
2		pmtrdifel.r	. . . . 5 ⊢ 𝑅 = ran (pmTrsp‘𝑁)
3		fveq2 6103	. . . . . . . . 9 ⊢ (𝑗 = 𝑛 → (𝑤‘𝑗) = (𝑤‘𝑛))
4	3	difeq1d 3689	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → ((𝑤‘𝑗) ∖ I ) = ((𝑤‘𝑛) ∖ I ))
5	4	dmeqd 5248	. . . . . . 7 ⊢ (𝑗 = 𝑛 → dom ((𝑤‘𝑗) ∖ I ) = dom ((𝑤‘𝑛) ∖ I ))
6	5	fveq2d 6107	. . . . . 6 ⊢ (𝑗 = 𝑛 → ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )) = ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑛) ∖ I )))
7	6	cbvmptv 4678	. . . . 5 ⊢ (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) = (𝑛 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑛) ∖ I )))
8	1, 2, 7	pmtrdifwrdellem1 17724	. . . 4 ⊢ (𝑤 ∈ Word 𝑇 → (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) ∈ Word 𝑅)
9	8	adantl 481	. . 3 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑤 ∈ Word 𝑇) → (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) ∈ Word 𝑅)
10	1, 2, 7	pmtrdifwrdellem2 17725	. . . 4 ⊢ (𝑤 ∈ Word 𝑇 → (#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))))
11	10	adantl 481	. . 3 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑤 ∈ Word 𝑇) → (#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))))
12	1, 2, 7	pmtrdifwrdel2lem1 17727	. . . . 5 ⊢ ((𝑤 ∈ Word 𝑇 ∧ 𝐾 ∈ 𝑁) → ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾)
13	12	ancoms 468	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑤 ∈ Word 𝑇) → ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾)
14	1, 2, 7	pmtrdifwrdellem3 17726	. . . . 5 ⊢ (𝑤 ∈ Word 𝑇 → ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥))
15	14	adantl 481	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑤 ∈ Word 𝑇) → ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥))
16		r19.26 3046	. . . 4 ⊢ (∀𝑖 ∈ (0..^(#‘𝑤))((((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥)) ↔ (∀𝑖 ∈ (0..^(#‘𝑤))(((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥)))
17	13, 15, 16	sylanbrc 695	. . 3 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑤 ∈ Word 𝑇) → ∀𝑖 ∈ (0..^(#‘𝑤))((((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥)))
18		fveq2 6103	. . . . . 6 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (#‘𝑢) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))))
19	18	eqeq2d 2620	. . . . 5 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → ((#‘𝑤) = (#‘𝑢) ↔ (#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))))))
20		fveq1 6102	. . . . . . . . 9 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (𝑢‘𝑖) = ((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖))
21	20	fveq1d 6105	. . . . . . . 8 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → ((𝑢‘𝑖)‘𝐾) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾))
22	21	eqeq1d 2612	. . . . . . 7 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (((𝑢‘𝑖)‘𝐾) = 𝐾 ↔ (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾))
23	20	fveq1d 6105	. . . . . . . . 9 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → ((𝑢‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥))
24	23	eqeq2d 2620	. . . . . . . 8 ⊢ (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)))
25	24	ralbidv 2969	. . . . . . 7 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥) ↔ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥)))
26	22, 25	anbi12d 743	. . . . . 6 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → ((((𝑢‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥)) ↔ ((((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥))))
27	26	ralbidv 2969	. . . . 5 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (∀𝑖 ∈ (0..^(#‘𝑤))(((𝑢‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥)) ↔ ∀𝑖 ∈ (0..^(#‘𝑤))((((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥))))
28	19, 27	anbi12d 743	. . . 4 ⊢ (𝑢 = (𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I ))) → (((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑢‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))) ↔ ((#‘𝑤) = (#‘(𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))((((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = (((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝑥)))))
29	28	rspcev 3282	. . 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))((((𝑗 ∈ (0..^(#‘𝑤)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑤‘𝑗) ∖ I )))‘𝑖)‘𝐾) = 𝐾 ∧ ∀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))(((𝑢‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))))
30	9, 11, 17, 29	syl12anc 1316	. 2 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑤 ∈ Word 𝑇) → ∃𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑢‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))))
31	30	ralrimiva 2949	1 ⊢ (𝐾 ∈ 𝑁 → ∀𝑤 ∈ Word 𝑇∃𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑢‘𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤‘𝑖)‘𝑥) = ((𝑢‘𝑖)‘𝑥))))

Syntax hints:   → wi 4   ∧ 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-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-pmtr 17685

This theorem is referenced by:  psgndiflemA  19766