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Mirrors > Home > MPE Home > Th. List > 1wlkdlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 1wlkd 27001. (Contributed by AV, 22-Jan-2021.) |
Ref | Expression |
---|---|
1wlkd.p | ⊢ 𝑃 = 〈“𝑋𝑌”〉 |
1wlkd.f | ⊢ 𝐹 = 〈“𝐽”〉 |
1wlkd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
1wlkd.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
1wlkdlem1 | ⊢ (𝜑 → 𝑃:(0...(#‘𝐹))⟶𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1wlkd.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | 1wlkd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
3 | 1, 2 | s2cld 13616 | . . 3 ⊢ (𝜑 → 〈“𝑋𝑌”〉 ∈ Word 𝑉) |
4 | wrdf 13310 | . . . 4 ⊢ (〈“𝑋𝑌”〉 ∈ Word 𝑉 → 〈“𝑋𝑌”〉:(0..^(#‘〈“𝑋𝑌”〉))⟶𝑉) | |
5 | 1z 11407 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
6 | fzval3 12536 | . . . . . . . 8 ⊢ (1 ∈ ℤ → (0...1) = (0..^(1 + 1))) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 ⊢ (0...1) = (0..^(1 + 1)) |
8 | 1wlkd.f | . . . . . . . . . 10 ⊢ 𝐹 = 〈“𝐽”〉 | |
9 | 8 | fveq2i 6194 | . . . . . . . . 9 ⊢ (#‘𝐹) = (#‘〈“𝐽”〉) |
10 | s1len 13385 | . . . . . . . . 9 ⊢ (#‘〈“𝐽”〉) = 1 | |
11 | 9, 10 | eqtri 2644 | . . . . . . . 8 ⊢ (#‘𝐹) = 1 |
12 | 11 | oveq2i 6661 | . . . . . . 7 ⊢ (0...(#‘𝐹)) = (0...1) |
13 | s2len 13634 | . . . . . . . . 9 ⊢ (#‘〈“𝑋𝑌”〉) = 2 | |
14 | df-2 11079 | . . . . . . . . 9 ⊢ 2 = (1 + 1) | |
15 | 13, 14 | eqtri 2644 | . . . . . . . 8 ⊢ (#‘〈“𝑋𝑌”〉) = (1 + 1) |
16 | 15 | oveq2i 6661 | . . . . . . 7 ⊢ (0..^(#‘〈“𝑋𝑌”〉)) = (0..^(1 + 1)) |
17 | 7, 12, 16 | 3eqtr4i 2654 | . . . . . 6 ⊢ (0...(#‘𝐹)) = (0..^(#‘〈“𝑋𝑌”〉)) |
18 | 17 | a1i 11 | . . . . 5 ⊢ (〈“𝑋𝑌”〉 ∈ Word 𝑉 → (0...(#‘𝐹)) = (0..^(#‘〈“𝑋𝑌”〉))) |
19 | 18 | feq2d 6031 | . . . 4 ⊢ (〈“𝑋𝑌”〉 ∈ Word 𝑉 → (〈“𝑋𝑌”〉:(0...(#‘𝐹))⟶𝑉 ↔ 〈“𝑋𝑌”〉:(0..^(#‘〈“𝑋𝑌”〉))⟶𝑉)) |
20 | 4, 19 | mpbird 247 | . . 3 ⊢ (〈“𝑋𝑌”〉 ∈ Word 𝑉 → 〈“𝑋𝑌”〉:(0...(#‘𝐹))⟶𝑉) |
21 | 3, 20 | syl 17 | . 2 ⊢ (𝜑 → 〈“𝑋𝑌”〉:(0...(#‘𝐹))⟶𝑉) |
22 | 1wlkd.p | . . 3 ⊢ 𝑃 = 〈“𝑋𝑌”〉 | |
23 | 22 | feq1i 6036 | . 2 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 〈“𝑋𝑌”〉:(0...(#‘𝐹))⟶𝑉) |
24 | 21, 23 | sylibr 224 | 1 ⊢ (𝜑 → 𝑃:(0...(#‘𝐹))⟶𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 2c2 11070 ℤcz 11377 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 〈“cs1 13294 〈“cs2 13586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 |
This theorem is referenced by: 1wlkd 27001 |
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