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Mirrors > Home > MPE Home > Th. List > clwlkf1clwwlklem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for clwlkf1clwwlklem 26376. (Contributed by Alexander van der Vekens, 5-Jul-2018.) |
Ref | Expression |
---|---|
clwlkfclwwlk.1 | ⊢ 𝐴 = (1st ‘𝑐) |
clwlkfclwwlk.2 | ⊢ 𝐵 = (2nd ‘𝑐) |
clwlkfclwwlk.c | ⊢ 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁} |
clwlkfclwwlk.f | ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr 〈0, (#‘𝐴)〉)) |
Ref | Expression |
---|---|
clwlkf1clwwlklem1 | ⊢ (𝑊 ∈ 𝐶 → 𝑁 ≤ (#‘(2nd ‘𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlkfclwwlk.1 | . . . . . 6 ⊢ 𝐴 = (1st ‘𝑐) | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝑐 = 𝑊 → (1st ‘𝑐) = (1st ‘𝑊)) | |
3 | 1, 2 | syl5eq 2656 | . . . . 5 ⊢ (𝑐 = 𝑊 → 𝐴 = (1st ‘𝑊)) |
4 | 3 | fveq2d 6107 | . . . 4 ⊢ (𝑐 = 𝑊 → (#‘𝐴) = (#‘(1st ‘𝑊))) |
5 | 4 | eqeq1d 2612 | . . 3 ⊢ (𝑐 = 𝑊 → ((#‘𝐴) = 𝑁 ↔ (#‘(1st ‘𝑊)) = 𝑁)) |
6 | clwlkfclwwlk.c | . . 3 ⊢ 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁} | |
7 | 5, 6 | elrab2 3333 | . 2 ⊢ (𝑊 ∈ 𝐶 ↔ (𝑊 ∈ (𝑉 ClWalks 𝐸) ∧ (#‘(1st ‘𝑊)) = 𝑁)) |
8 | eqid 2610 | . . . . 5 ⊢ (1st ‘𝑊) = (1st ‘𝑊) | |
9 | eqid 2610 | . . . . 5 ⊢ (2nd ‘𝑊) = (2nd ‘𝑊) | |
10 | 8, 9 | clwlkcompim 26292 | . . . 4 ⊢ (𝑊 ∈ (𝑉 ClWalks 𝐸) → (((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑖)) = {((2nd ‘𝑊)‘𝑖), ((2nd ‘𝑊)‘(𝑖 + 1))} ∧ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(#‘(1st ‘𝑊)))))) |
11 | lencl 13179 | . . . . . 6 ⊢ ((1st ‘𝑊) ∈ Word dom 𝐸 → (#‘(1st ‘𝑊)) ∈ ℕ0) | |
12 | ffn 5958 | . . . . . 6 ⊢ ((2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉 → (2nd ‘𝑊) Fn (0...(#‘(1st ‘𝑊)))) | |
13 | nn0re 11178 | . . . . . . . . . 10 ⊢ ((#‘(1st ‘𝑊)) ∈ ℕ0 → (#‘(1st ‘𝑊)) ∈ ℝ) | |
14 | 13 | lep1d 10834 | . . . . . . . . 9 ⊢ ((#‘(1st ‘𝑊)) ∈ ℕ0 → (#‘(1st ‘𝑊)) ≤ ((#‘(1st ‘𝑊)) + 1)) |
15 | 14 | adantr 480 | . . . . . . . 8 ⊢ (((#‘(1st ‘𝑊)) ∈ ℕ0 ∧ (2nd ‘𝑊) Fn (0...(#‘(1st ‘𝑊)))) → (#‘(1st ‘𝑊)) ≤ ((#‘(1st ‘𝑊)) + 1)) |
16 | fnfz0hash 13087 | . . . . . . . 8 ⊢ (((#‘(1st ‘𝑊)) ∈ ℕ0 ∧ (2nd ‘𝑊) Fn (0...(#‘(1st ‘𝑊)))) → (#‘(2nd ‘𝑊)) = ((#‘(1st ‘𝑊)) + 1)) | |
17 | 15, 16 | breqtrrd 4611 | . . . . . . 7 ⊢ (((#‘(1st ‘𝑊)) ∈ ℕ0 ∧ (2nd ‘𝑊) Fn (0...(#‘(1st ‘𝑊)))) → (#‘(1st ‘𝑊)) ≤ (#‘(2nd ‘𝑊))) |
18 | breq1 4586 | . . . . . . 7 ⊢ ((#‘(1st ‘𝑊)) = 𝑁 → ((#‘(1st ‘𝑊)) ≤ (#‘(2nd ‘𝑊)) ↔ 𝑁 ≤ (#‘(2nd ‘𝑊)))) | |
19 | 17, 18 | syl5ibcom 234 | . . . . . 6 ⊢ (((#‘(1st ‘𝑊)) ∈ ℕ0 ∧ (2nd ‘𝑊) Fn (0...(#‘(1st ‘𝑊)))) → ((#‘(1st ‘𝑊)) = 𝑁 → 𝑁 ≤ (#‘(2nd ‘𝑊)))) |
20 | 11, 12, 19 | syl2an 493 | . . . . 5 ⊢ (((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉) → ((#‘(1st ‘𝑊)) = 𝑁 → 𝑁 ≤ (#‘(2nd ‘𝑊)))) |
21 | 20 | adantr 480 | . . . 4 ⊢ ((((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑖)) = {((2nd ‘𝑊)‘𝑖), ((2nd ‘𝑊)‘(𝑖 + 1))} ∧ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(#‘(1st ‘𝑊))))) → ((#‘(1st ‘𝑊)) = 𝑁 → 𝑁 ≤ (#‘(2nd ‘𝑊)))) |
22 | 10, 21 | syl 17 | . . 3 ⊢ (𝑊 ∈ (𝑉 ClWalks 𝐸) → ((#‘(1st ‘𝑊)) = 𝑁 → 𝑁 ≤ (#‘(2nd ‘𝑊)))) |
23 | 22 | imp 444 | . 2 ⊢ ((𝑊 ∈ (𝑉 ClWalks 𝐸) ∧ (#‘(1st ‘𝑊)) = 𝑁) → 𝑁 ≤ (#‘(2nd ‘𝑊))) |
24 | 7, 23 | sylbi 206 | 1 ⊢ (𝑊 ∈ 𝐶 → 𝑁 ≤ (#‘(2nd ‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 {cpr 4127 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 ℕ0cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 substr csubstr 13150 ClWalks cclwlk 26275 |
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-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 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-wlk 26036 df-clwlk 26278 |
This theorem is referenced by: clwlkf1clwwlklem 26376 |
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