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Mirrors > Home > MPE Home > Th. List > erclwwlkeqlen | Structured version Visualization version GIF version |
Description: If two classes are equivalent regarding ∼, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.) |
Ref | Expression |
---|---|
erclwwlk.r | ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} |
Ref | Expression |
---|---|
erclwwlkeqlen | ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 → (#‘𝑈) = (#‘𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erclwwlk.r | . . 3 ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} | |
2 | 1 | erclwwlkeq 26339 | . 2 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))) |
3 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘(𝑊 cyclShift 𝑛))) | |
4 | clwwlkprop 26298 | . . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉)) | |
5 | pm3.2 462 | . . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑉 → (𝑛 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊))))) | |
6 | 5 | 3ad2ant3 1077 | . . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → (𝑛 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊))))) |
7 | 4, 6 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑛 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊))))) |
8 | 7 | ad2antlr 759 | . . . . . . . . . . 11 ⊢ (((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (𝑛 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊))))) |
9 | 8 | imp 444 | . . . . . . . . . 10 ⊢ ((((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊)))) |
10 | elfzelz 12213 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ (0...(#‘𝑊)) → 𝑛 ∈ ℤ) | |
11 | cshwlen 13396 | . . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊)) | |
12 | 10, 11 | sylan2 490 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊)) |
13 | 9, 12 | syl 17 | . . . . . . . . 9 ⊢ ((((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊)) |
14 | 3, 13 | sylan9eqr 2666 | . . . . . . . 8 ⊢ (((((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) ∧ 𝑈 = (𝑊 cyclShift 𝑛)) → (#‘𝑈) = (#‘𝑊)) |
15 | 14 | ex 449 | . . . . . . 7 ⊢ ((((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊))) |
16 | 15 | rexlimdva 3013 | . . . . . 6 ⊢ (((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊))) |
17 | 16 | ex 449 | . . . . 5 ⊢ ((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) → ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊)))) |
18 | 17 | com23 84 | . . . 4 ⊢ ((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (#‘𝑈) = (#‘𝑊)))) |
19 | 18 | 3impia 1253 | . . 3 ⊢ ((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)) → ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (#‘𝑈) = (#‘𝑊))) |
20 | 19 | com12 32 | . 2 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → ((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)) → (#‘𝑈) = (#‘𝑊))) |
21 | 2, 20 | sylbid 229 | 1 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 → (#‘𝑈) = (#‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 {copab 4642 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℤcz 11254 ...cfz 12197 #chash 12979 Word cword 13146 cyclShift ccsh 13385 ClWWalks cclwwlk 26276 |
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 ax-pre-sup 9893 |
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-rmo 2904 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-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 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-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 df-csh 13386 df-clwwlk 26279 |
This theorem is referenced by: erclwwlksym 26342 erclwwlktr 26343 |
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