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Mirrors > Home > MPE Home > Th. List > 1wlkdlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for 1wlkd 28221. (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 14433 | . . 3 ⊢ (𝜑 → 〈“𝑋𝑌”〉 ∈ Word 𝑉) |
4 | wrdf 14071 | . . . 4 ⊢ (〈“𝑋𝑌”〉 ∈ Word 𝑉 → 〈“𝑋𝑌”〉:(0..^(♯‘〈“𝑋𝑌”〉))⟶𝑉) | |
5 | 1z 12204 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
6 | fzval3 13308 | . . . . . . . 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 6717 | . . . . . . . . 9 ⊢ (♯‘𝐹) = (♯‘〈“𝐽”〉) |
10 | s1len 14160 | . . . . . . . . 9 ⊢ (♯‘〈“𝐽”〉) = 1 | |
11 | 9, 10 | eqtri 2765 | . . . . . . . 8 ⊢ (♯‘𝐹) = 1 |
12 | 11 | oveq2i 7221 | . . . . . . 7 ⊢ (0...(♯‘𝐹)) = (0...1) |
13 | s2len 14451 | . . . . . . . . 9 ⊢ (♯‘〈“𝑋𝑌”〉) = 2 | |
14 | df-2 11890 | . . . . . . . . 9 ⊢ 2 = (1 + 1) | |
15 | 13, 14 | eqtri 2765 | . . . . . . . 8 ⊢ (♯‘〈“𝑋𝑌”〉) = (1 + 1) |
16 | 15 | oveq2i 7221 | . . . . . . 7 ⊢ (0..^(♯‘〈“𝑋𝑌”〉)) = (0..^(1 + 1)) |
17 | 7, 12, 16 | 3eqtr4i 2775 | . . . . . 6 ⊢ (0...(♯‘𝐹)) = (0..^(♯‘〈“𝑋𝑌”〉)) |
18 | 17 | a1i 11 | . . . . 5 ⊢ (〈“𝑋𝑌”〉 ∈ Word 𝑉 → (0...(♯‘𝐹)) = (0..^(♯‘〈“𝑋𝑌”〉))) |
19 | 18 | feq2d 6528 | . . . 4 ⊢ (〈“𝑋𝑌”〉 ∈ Word 𝑉 → (〈“𝑋𝑌”〉:(0...(♯‘𝐹))⟶𝑉 ↔ 〈“𝑋𝑌”〉:(0..^(♯‘〈“𝑋𝑌”〉))⟶𝑉)) |
20 | 4, 19 | mpbird 260 | . . 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 6533 | . 2 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 〈“𝑋𝑌”〉:(0...(♯‘𝐹))⟶𝑉) |
24 | 21, 23 | sylibr 237 | 1 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ⟶wf 6373 ‘cfv 6377 (class class class)co 7210 0cc0 10726 1c1 10727 + caddc 10729 2c2 11882 ℤcz 12173 ...cfz 13092 ..^cfzo 13235 ♯chash 13893 Word cword 14066 〈“cs1 14149 〈“cs2 14403 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-card 9552 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-2 11890 df-n0 12088 df-z 12174 df-uz 12436 df-fz 13093 df-fzo 13236 df-hash 13894 df-word 14067 df-concat 14123 df-s1 14150 df-s2 14410 |
This theorem is referenced by: 1wlkd 28221 |
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