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Mirrors > Home > MPE Home > Th. List > cats1fv | Structured version Visualization version GIF version |
Description: A symbol other than the last in a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
cats1cld.1 | ⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩) |
cats1cli.2 | ⊢ 𝑆 ∈ Word V |
cats1fvn.3 | ⊢ (♯‘𝑆) = 𝑀 |
cats1fv.4 | ⊢ (𝑌 ∈ 𝑉 → (𝑆‘𝑁) = 𝑌) |
cats1fv.5 | ⊢ 𝑁 ∈ ℕ0 |
cats1fv.6 | ⊢ 𝑁 < 𝑀 |
Ref | Expression |
---|---|
cats1fv | ⊢ (𝑌 ∈ 𝑉 → (𝑇‘𝑁) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cats1cld.1 | . . . 4 ⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩) | |
2 | 1 | fveq1i 6885 | . . 3 ⊢ (𝑇‘𝑁) = ((𝑆 ++ ⟨“𝑋”⟩)‘𝑁) |
3 | cats1cli.2 | . . . 4 ⊢ 𝑆 ∈ Word V | |
4 | s1cli 14558 | . . . 4 ⊢ ⟨“𝑋”⟩ ∈ Word V | |
5 | cats1fv.5 | . . . . . 6 ⊢ 𝑁 ∈ ℕ0 | |
6 | nn0uz 12865 | . . . . . 6 ⊢ ℕ0 = (ℤ≥‘0) | |
7 | 5, 6 | eleqtri 2825 | . . . . 5 ⊢ 𝑁 ∈ (ℤ≥‘0) |
8 | lencl 14486 | . . . . . 6 ⊢ (𝑆 ∈ Word V → (♯‘𝑆) ∈ ℕ0) | |
9 | nn0z 12584 | . . . . . 6 ⊢ ((♯‘𝑆) ∈ ℕ0 → (♯‘𝑆) ∈ ℤ) | |
10 | 3, 8, 9 | mp2b 10 | . . . . 5 ⊢ (♯‘𝑆) ∈ ℤ |
11 | cats1fv.6 | . . . . . 6 ⊢ 𝑁 < 𝑀 | |
12 | cats1fvn.3 | . . . . . 6 ⊢ (♯‘𝑆) = 𝑀 | |
13 | 11, 12 | breqtrri 5168 | . . . . 5 ⊢ 𝑁 < (♯‘𝑆) |
14 | elfzo2 13638 | . . . . 5 ⊢ (𝑁 ∈ (0..^(♯‘𝑆)) ↔ (𝑁 ∈ (ℤ≥‘0) ∧ (♯‘𝑆) ∈ ℤ ∧ 𝑁 < (♯‘𝑆))) | |
15 | 7, 10, 13, 14 | mpbir3an 1338 | . . . 4 ⊢ 𝑁 ∈ (0..^(♯‘𝑆)) |
16 | ccatval1 14530 | . . . 4 ⊢ ((𝑆 ∈ Word V ∧ ⟨“𝑋”⟩ ∈ Word V ∧ 𝑁 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ ⟨“𝑋”⟩)‘𝑁) = (𝑆‘𝑁)) | |
17 | 3, 4, 15, 16 | mp3an 1457 | . . 3 ⊢ ((𝑆 ++ ⟨“𝑋”⟩)‘𝑁) = (𝑆‘𝑁) |
18 | 2, 17 | eqtri 2754 | . 2 ⊢ (𝑇‘𝑁) = (𝑆‘𝑁) |
19 | cats1fv.4 | . 2 ⊢ (𝑌 ∈ 𝑉 → (𝑆‘𝑁) = 𝑌) | |
20 | 18, 19 | eqtrid 2778 | 1 ⊢ (𝑌 ∈ 𝑉 → (𝑇‘𝑁) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 0cc0 11109 < clt 11249 ℕ0cn0 12473 ℤcz 12559 ℤ≥cuz 12823 ..^cfzo 13630 ♯chash 14292 Word cword 14467 ++ cconcat 14523 ⟨“cs1 14548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14293 df-word 14468 df-concat 14524 df-s1 14549 |
This theorem is referenced by: s2fv0 14841 s3fv0 14845 s3fv1 14846 s4fv0 14849 s4fv1 14850 s4fv2 14851 |
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