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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 6907 | . . 3 ⊢ (𝑇‘𝑁) = ((𝑆 ++ 〈“𝑋”〉)‘𝑁) | 
| 3 | cats1cli.2 | . . . 4 ⊢ 𝑆 ∈ Word V | |
| 4 | s1cli 14643 | . . . 4 ⊢ 〈“𝑋”〉 ∈ Word V | |
| 5 | cats1fv.5 | . . . . . 6 ⊢ 𝑁 ∈ ℕ0 | |
| 6 | nn0uz 12920 | . . . . . 6 ⊢ ℕ0 = (ℤ≥‘0) | |
| 7 | 5, 6 | eleqtri 2839 | . . . . 5 ⊢ 𝑁 ∈ (ℤ≥‘0) | 
| 8 | lencl 14571 | . . . . . 6 ⊢ (𝑆 ∈ Word V → (♯‘𝑆) ∈ ℕ0) | |
| 9 | nn0z 12638 | . . . . . 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 5170 | . . . . 5 ⊢ 𝑁 < (♯‘𝑆) | 
| 14 | elfzo2 13702 | . . . . 5 ⊢ (𝑁 ∈ (0..^(♯‘𝑆)) ↔ (𝑁 ∈ (ℤ≥‘0) ∧ (♯‘𝑆) ∈ ℤ ∧ 𝑁 < (♯‘𝑆))) | |
| 15 | 7, 10, 13, 14 | mpbir3an 1342 | . . . 4 ⊢ 𝑁 ∈ (0..^(♯‘𝑆)) | 
| 16 | ccatval1 14615 | . . . 4 ⊢ ((𝑆 ∈ Word V ∧ 〈“𝑋”〉 ∈ Word V ∧ 𝑁 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 〈“𝑋”〉)‘𝑁) = (𝑆‘𝑁)) | |
| 17 | 3, 4, 15, 16 | mp3an 1463 | . . 3 ⊢ ((𝑆 ++ 〈“𝑋”〉)‘𝑁) = (𝑆‘𝑁) | 
| 18 | 2, 17 | eqtri 2765 | . 2 ⊢ (𝑇‘𝑁) = (𝑆‘𝑁) | 
| 19 | cats1fv.4 | . 2 ⊢ (𝑌 ∈ 𝑉 → (𝑆‘𝑁) = 𝑌) | |
| 20 | 18, 19 | eqtrid 2789 | 1 ⊢ (𝑌 ∈ 𝑉 → (𝑇‘𝑁) = 𝑌) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 < clt 11295 ℕ0cn0 12526 ℤcz 12613 ℤ≥cuz 12878 ..^cfzo 13694 ♯chash 14369 Word cword 14552 ++ cconcat 14608 〈“cs1 14633 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 | 
| This theorem is referenced by: s2fv0 14926 s3fv0 14930 s3fv1 14931 s4fv0 14934 s4fv1 14935 s4fv2 14936 | 
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