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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 1133 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 𝑊 ∈ Word 𝑉) | |
2 | s1cl 14558 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉) | |
3 | 2 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ⟨“𝑆”⟩ ∈ Word 𝑉) |
4 | lencl 14489 | . . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
5 | 4 | nn0zd 12588 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℤ) |
6 | elfzomin 13710 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℤ → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + 1))) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + 1))) |
8 | s1len 14562 | . . . . . . . . 9 ⊢ (♯‘⟨“𝑆”⟩) = 1 | |
9 | 8 | oveq2i 7416 | . . . . . . . 8 ⊢ ((♯‘𝑊) + (♯‘⟨“𝑆”⟩)) = ((♯‘𝑊) + 1) |
10 | 9 | oveq2i 7416 | . . . . . . 7 ⊢ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩))) = ((♯‘𝑊)..^((♯‘𝑊) + 1)) |
11 | 7, 10 | eleqtrrdi 2838 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩)))) |
12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩)))) |
13 | eleq1 2815 | . . . . . 6 ⊢ (𝐼 = (♯‘𝑊) → (𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩))) ↔ (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩))))) | |
14 | 13 | adantl 481 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩))) ↔ (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩))))) |
15 | 12, 14 | mpbird 257 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩)))) |
16 | 15 | 3adant2 1128 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩)))) |
17 | ccatval2 14534 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘⟨“𝑆”⟩)))) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = (⟨“𝑆”⟩‘(𝐼 − (♯‘𝑊)))) | |
18 | 1, 3, 16, 17 | syl3anc 1368 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = (⟨“𝑆”⟩‘(𝐼 − (♯‘𝑊)))) |
19 | oveq1 7412 | . . . . 5 ⊢ (𝐼 = (♯‘𝑊) → (𝐼 − (♯‘𝑊)) = ((♯‘𝑊) − (♯‘𝑊))) | |
20 | 19 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (𝐼 − (♯‘𝑊)) = ((♯‘𝑊) − (♯‘𝑊))) |
21 | 4 | nn0cnd 12538 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℂ) |
22 | 21 | subidd 11563 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) − (♯‘𝑊)) = 0) |
23 | 22 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((♯‘𝑊) − (♯‘𝑊)) = 0) |
24 | 20, 23 | eqtrd 2766 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (𝐼 − (♯‘𝑊)) = 0) |
25 | 24 | fveq2d 6889 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (⟨“𝑆”⟩‘(𝐼 − (♯‘𝑊))) = (⟨“𝑆”⟩‘0)) |
26 | s1fv 14566 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (⟨“𝑆”⟩‘0) = 𝑆) | |
27 | 26 | 3ad2ant2 1131 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (⟨“𝑆”⟩‘0) = 𝑆) |
28 | 18, 25, 27 | 3eqtrd 2770 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 + caddc 11115 − cmin 11448 ℤcz 12562 ..^cfzo 13633 ♯chash 14295 Word cword 14470 ++ cconcat 14526 ⟨“cs1 14551 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-concat 14527 df-s1 14552 |
This theorem is referenced by: ccatws1ls 14589 ccatw2s1p1 14592 ccatw2s1p2 14593 gsmsymgrfixlem1 19347 gsmsymgreqlem2 19351 wwlksnext 29656 clwwlkwwlksb 29816 clwwlknonwwlknonb 29868 |
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