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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 6903 | . . 3 ⊢ (𝑇‘𝑁) = ((𝑆 ++ ⟨“𝑋”⟩)‘𝑁) |
3 | cats1cli.2 | . . . 4 ⊢ 𝑆 ∈ Word V | |
4 | s1cli 14595 | . . . 4 ⊢ ⟨“𝑋”⟩ ∈ Word V | |
5 | cats1fv.5 | . . . . . 6 ⊢ 𝑁 ∈ ℕ0 | |
6 | nn0uz 12902 | . . . . . 6 ⊢ ℕ0 = (ℤ≥‘0) | |
7 | 5, 6 | eleqtri 2827 | . . . . 5 ⊢ 𝑁 ∈ (ℤ≥‘0) |
8 | lencl 14523 | . . . . . 6 ⊢ (𝑆 ∈ Word V → (♯‘𝑆) ∈ ℕ0) | |
9 | nn0z 12621 | . . . . . 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 5179 | . . . . 5 ⊢ 𝑁 < (♯‘𝑆) |
14 | elfzo2 13675 | . . . . 5 ⊢ (𝑁 ∈ (0..^(♯‘𝑆)) ↔ (𝑁 ∈ (ℤ≥‘0) ∧ (♯‘𝑆) ∈ ℤ ∧ 𝑁 < (♯‘𝑆))) | |
15 | 7, 10, 13, 14 | mpbir3an 1338 | . . . 4 ⊢ 𝑁 ∈ (0..^(♯‘𝑆)) |
16 | ccatval1 14567 | . . . 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 2756 | . 2 ⊢ (𝑇‘𝑁) = (𝑆‘𝑁) |
19 | cats1fv.4 | . 2 ⊢ (𝑌 ∈ 𝑉 → (𝑆‘𝑁) = 𝑌) | |
20 | 18, 19 | eqtrid 2780 | 1 ⊢ (𝑌 ∈ 𝑉 → (𝑇‘𝑁) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 0cc0 11146 < clt 11286 ℕ0cn0 12510 ℤcz 12596 ℤ≥cuz 12860 ..^cfzo 13667 ♯chash 14329 Word cword 14504 ++ cconcat 14560 ⟨“cs1 14585 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-concat 14561 df-s1 14586 |
This theorem is referenced by: s2fv0 14878 s3fv0 14882 s3fv1 14883 s4fv0 14886 s4fv1 14887 s4fv2 14888 |
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