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Mirrors > Home > MPE Home > Th. List > ccats1val2 | Structured version Visualization version GIF version |
Description: Value of the symbol concatenated with a word. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Proof shortened by Alexander van der Vekens, 14-Oct-2018.) |
Ref | Expression |
---|---|
ccats1val2 | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1117 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 𝑊 ∈ Word 𝑉) | |
2 | s1cl 13764 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 〈“𝑆”〉 ∈ Word 𝑉) | |
3 | 2 | 3ad2ant2 1115 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 〈“𝑆”〉 ∈ Word 𝑉) |
4 | lencl 13693 | . . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
5 | 4 | nn0zd 11897 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℤ) |
6 | elfzomin 12923 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℤ → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + 1))) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + 1))) |
8 | s1len 13768 | . . . . . . . . 9 ⊢ (♯‘〈“𝑆”〉) = 1 | |
9 | 8 | oveq2i 6986 | . . . . . . . 8 ⊢ ((♯‘𝑊) + (♯‘〈“𝑆”〉)) = ((♯‘𝑊) + 1) |
10 | 9 | oveq2i 6986 | . . . . . . 7 ⊢ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉))) = ((♯‘𝑊)..^((♯‘𝑊) + 1)) |
11 | 7, 10 | syl6eleqr 2872 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉)))) |
12 | 11 | adantr 473 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉)))) |
13 | eleq1 2848 | . . . . . 6 ⊢ (𝐼 = (♯‘𝑊) → (𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉))) ↔ (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉))))) | |
14 | 13 | adantl 474 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉))) ↔ (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉))))) |
15 | 12, 14 | mpbird 249 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉)))) |
16 | 15 | 3adant2 1112 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉)))) |
17 | ccatval2 13740 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉)))) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = (〈“𝑆”〉‘(𝐼 − (♯‘𝑊)))) | |
18 | 1, 3, 16, 17 | syl3anc 1352 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = (〈“𝑆”〉‘(𝐼 − (♯‘𝑊)))) |
19 | oveq1 6982 | . . . . 5 ⊢ (𝐼 = (♯‘𝑊) → (𝐼 − (♯‘𝑊)) = ((♯‘𝑊) − (♯‘𝑊))) | |
20 | 19 | 3ad2ant3 1116 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (𝐼 − (♯‘𝑊)) = ((♯‘𝑊) − (♯‘𝑊))) |
21 | 4 | nn0cnd 11768 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℂ) |
22 | 21 | subidd 10785 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) − (♯‘𝑊)) = 0) |
23 | 22 | 3ad2ant1 1114 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((♯‘𝑊) − (♯‘𝑊)) = 0) |
24 | 20, 23 | eqtrd 2809 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (𝐼 − (♯‘𝑊)) = 0) |
25 | 24 | fveq2d 6501 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (〈“𝑆”〉‘(𝐼 − (♯‘𝑊))) = (〈“𝑆”〉‘0)) |
26 | s1fv 13772 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉‘0) = 𝑆) | |
27 | 26 | 3ad2ant2 1115 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (〈“𝑆”〉‘0) = 𝑆) |
28 | 18, 25, 27 | 3eqtrd 2813 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ‘cfv 6186 (class class class)co 6975 0cc0 10334 1c1 10335 + caddc 10337 − cmin 10669 ℤcz 11792 ..^cfzo 12848 ♯chash 13504 Word cword 13671 ++ cconcat 13732 〈“cs1 13757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-oadd 7908 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-n0 11707 df-z 11793 df-uz 12058 df-fz 12708 df-fzo 12849 df-hash 13505 df-word 13672 df-concat 13733 df-s1 13758 |
This theorem is referenced by: ccatws1ls 13795 ccatw2s1p1 13798 ccatw2s1p2 13799 gsmsymgrfixlem1 18329 gsmsymgreqlem2 18333 wwlksnext 27397 clwwlkwwlksb 27593 clwwlknonwwlknonb 27650 |
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