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Mirrors > Home > MPE Home > Th. List > pmtrdifwrdellem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for pmtrdifwrdel 19434. (Contributed by AV, 15-Jan-2019.) |
Ref | Expression |
---|---|
pmtrdifel.t | ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) |
pmtrdifel.r | ⊢ 𝑅 = ran (pmTrsp‘𝑁) |
pmtrdifwrdel.0 | ⊢ 𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑥) ∖ I ))) |
Ref | Expression |
---|---|
pmtrdifwrdellem3 | ⊢ (𝑊 ∈ Word 𝑇 → ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdsymbcl 14504 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑖) ∈ 𝑇) | |
2 | pmtrdifel.t | . . . . 5 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) | |
3 | pmtrdifel.r | . . . . 5 ⊢ 𝑅 = ran (pmTrsp‘𝑁) | |
4 | eqid 2728 | . . . . 5 ⊢ ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I )) = ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I )) | |
5 | 2, 3, 4 | pmtrdifellem3 19427 | . . . 4 ⊢ ((𝑊‘𝑖) ∈ 𝑇 → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = (((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))‘𝑛)) |
6 | 1, 5 | syl 17 | . . 3 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = (((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))‘𝑛)) |
7 | pmtrdifwrdel.0 | . . . . . . 7 ⊢ 𝑈 = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑥) ∖ I ))) | |
8 | fveq2 6892 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑖 → (𝑊‘𝑥) = (𝑊‘𝑖)) | |
9 | 8 | difeq1d 4118 | . . . . . . . . 9 ⊢ (𝑥 = 𝑖 → ((𝑊‘𝑥) ∖ I ) = ((𝑊‘𝑖) ∖ I )) |
10 | 9 | dmeqd 5903 | . . . . . . . 8 ⊢ (𝑥 = 𝑖 → dom ((𝑊‘𝑥) ∖ I ) = dom ((𝑊‘𝑖) ∖ I )) |
11 | 10 | fveq2d 6896 | . . . . . . 7 ⊢ (𝑥 = 𝑖 → ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑥) ∖ I )) = ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))) |
12 | simpr 484 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → 𝑖 ∈ (0..^(♯‘𝑊))) | |
13 | fvexd 6907 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I )) ∈ V) | |
14 | 7, 11, 12, 13 | fvmptd3 7023 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → (𝑈‘𝑖) = ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))) |
15 | 14 | fveq1d 6894 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ((𝑈‘𝑖)‘𝑛) = (((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))‘𝑛)) |
16 | 15 | eqeq2d 2739 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → (((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) ↔ ((𝑊‘𝑖)‘𝑛) = (((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))‘𝑛))) |
17 | 16 | ralbidv 3173 | . . 3 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → (∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = (((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))‘𝑛))) |
18 | 6, 17 | mpbird 257 | . 2 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) |
19 | 18 | ralrimiva 3142 | 1 ⊢ (𝑊 ∈ Word 𝑇 → ∀𝑖 ∈ (0..^(♯‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 ∖ cdif 3942 {csn 4625 ↦ cmpt 5226 I cid 5570 dom cdm 5673 ran crn 5674 ‘cfv 6543 (class class class)co 7415 0cc0 11133 ..^cfzo 13654 ♯chash 14316 Word cword 14491 pmTrspcpmtr 19390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-pmtr 19391 |
This theorem is referenced by: pmtrdifwrdel 19434 pmtrdifwrdel2 19435 |
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