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Mirrors > Home > MPE Home > Th. List > pmtrdifwrdellem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for pmtrdifwrdel 18909. (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 14114 | . . . 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 2739 | . . . . 5 ⊢ ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I )) = ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I )) | |
5 | 2, 3, 4 | pmtrdifellem3 18902 | . . . 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 6738 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑖 → (𝑊‘𝑥) = (𝑊‘𝑖)) | |
9 | 8 | difeq1d 4052 | . . . . . . . . 9 ⊢ (𝑥 = 𝑖 → ((𝑊‘𝑥) ∖ I ) = ((𝑊‘𝑖) ∖ I )) |
10 | 9 | dmeqd 5791 | . . . . . . . 8 ⊢ (𝑥 = 𝑖 → dom ((𝑊‘𝑥) ∖ I ) = dom ((𝑊‘𝑖) ∖ I )) |
11 | 10 | fveq2d 6742 | . . . . . . 7 ⊢ (𝑥 = 𝑖 → ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑥) ∖ I )) = ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))) |
12 | simpr 488 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → 𝑖 ∈ (0..^(♯‘𝑊))) | |
13 | fvexd 6753 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I )) ∈ V) | |
14 | 7, 11, 12, 13 | fvmptd3 6862 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → (𝑈‘𝑖) = ((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))) |
15 | 14 | fveq1d 6740 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ((𝑈‘𝑖)‘𝑛) = (((pmTrsp‘𝑁)‘dom ((𝑊‘𝑖) ∖ I ))‘𝑛)) |
16 | 15 | eqeq2d 2750 | . . . 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 3121 | . . 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 260 | . 2 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊‘𝑖)‘𝑛) = ((𝑈‘𝑖)‘𝑛)) |
19 | 18 | ralrimiva 3108 | 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 399 = wceq 1543 ∈ wcel 2112 ∀wral 3064 Vcvv 3423 ∖ cdif 3880 {csn 4557 ↦ cmpt 5151 I cid 5470 dom cdm 5568 ran crn 5569 ‘cfv 6400 (class class class)co 7234 0cc0 10758 ..^cfzo 13267 ♯chash 13928 Word cword 14101 pmTrspcpmtr 18865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-int 4876 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-1st 7782 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-1o 8225 df-2o 8226 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-card 9584 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-nn 11860 df-n0 12120 df-z 12206 df-uz 12468 df-fz 13125 df-fzo 13268 df-hash 13929 df-word 14102 df-pmtr 18866 |
This theorem is referenced by: pmtrdifwrdel 18909 pmtrdifwrdel2 18910 |
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