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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 1135 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 𝑊 ∈ Word 𝑉) | |
2 | s1cl 14637 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 〈“𝑆”〉 ∈ Word 𝑉) | |
3 | 2 | 3ad2ant2 1133 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 〈“𝑆”〉 ∈ Word 𝑉) |
4 | lencl 14568 | . . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
5 | 4 | nn0zd 12637 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℤ) |
6 | elfzomin 13773 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℤ → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + 1))) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + 1))) |
8 | s1len 14641 | . . . . . . . . 9 ⊢ (♯‘〈“𝑆”〉) = 1 | |
9 | 8 | oveq2i 7442 | . . . . . . . 8 ⊢ ((♯‘𝑊) + (♯‘〈“𝑆”〉)) = ((♯‘𝑊) + 1) |
10 | 9 | oveq2i 7442 | . . . . . . 7 ⊢ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉))) = ((♯‘𝑊)..^((♯‘𝑊) + 1)) |
11 | 7, 10 | eleqtrrdi 2850 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉)))) |
12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (♯‘𝑊) ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉)))) |
13 | eleq1 2827 | . . . . . 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 1130 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → 𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉)))) |
17 | ccatval2 14613 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝑊)..^((♯‘𝑊) + (♯‘〈“𝑆”〉)))) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = (〈“𝑆”〉‘(𝐼 − (♯‘𝑊)))) | |
18 | 1, 3, 16, 17 | syl3anc 1370 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = (〈“𝑆”〉‘(𝐼 − (♯‘𝑊)))) |
19 | oveq1 7438 | . . . . 5 ⊢ (𝐼 = (♯‘𝑊) → (𝐼 − (♯‘𝑊)) = ((♯‘𝑊) − (♯‘𝑊))) | |
20 | 19 | 3ad2ant3 1134 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (𝐼 − (♯‘𝑊)) = ((♯‘𝑊) − (♯‘𝑊))) |
21 | 4 | nn0cnd 12587 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℂ) |
22 | 21 | subidd 11606 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) − (♯‘𝑊)) = 0) |
23 | 22 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((♯‘𝑊) − (♯‘𝑊)) = 0) |
24 | 20, 23 | eqtrd 2775 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (𝐼 − (♯‘𝑊)) = 0) |
25 | 24 | fveq2d 6911 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (〈“𝑆”〉‘(𝐼 − (♯‘𝑊))) = (〈“𝑆”〉‘0)) |
26 | s1fv 14645 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉‘0) = 𝑆) | |
27 | 26 | 3ad2ant2 1133 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → (〈“𝑆”〉‘0) = 𝑆) |
28 | 18, 25, 27 | 3eqtrd 2779 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (♯‘𝑊)) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 − cmin 11490 ℤcz 12611 ..^cfzo 13691 ♯chash 14366 Word cword 14549 ++ cconcat 14605 〈“cs1 14630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 |
This theorem is referenced by: ccatws1ls 14668 ccatw2s1p1 14671 ccatw2s1p2 14672 gsmsymgrfixlem1 19460 gsmsymgreqlem2 19464 wwlksnext 29923 clwwlkwwlksb 30083 clwwlknonwwlknonb 30135 ccatws1f1olast 32922 chnind 32985 |
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