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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 1054 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → 𝑊 ∈ Word 𝑉) | |
2 | s1cl 13235 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 〈“𝑆”〉 ∈ Word 𝑉) | |
3 | 2 | 3ad2ant2 1076 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → 〈“𝑆”〉 ∈ Word 𝑉) |
4 | lencl 13179 | . . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0) | |
5 | 4 | nn0zd 11356 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℤ) |
6 | elfzomin 12406 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℤ → (#‘𝑊) ∈ ((#‘𝑊)..^((#‘𝑊) + 1))) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ((#‘𝑊)..^((#‘𝑊) + 1))) |
8 | s1len 13238 | . . . . . . . . 9 ⊢ (#‘〈“𝑆”〉) = 1 | |
9 | 8 | oveq2i 6560 | . . . . . . . 8 ⊢ ((#‘𝑊) + (#‘〈“𝑆”〉)) = ((#‘𝑊) + 1) |
10 | 9 | oveq2i 6560 | . . . . . . 7 ⊢ ((#‘𝑊)..^((#‘𝑊) + (#‘〈“𝑆”〉))) = ((#‘𝑊)..^((#‘𝑊) + 1)) |
11 | 7, 10 | syl6eleqr 2699 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ((#‘𝑊)..^((#‘𝑊) + (#‘〈“𝑆”〉)))) |
12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 = (#‘𝑊)) → (#‘𝑊) ∈ ((#‘𝑊)..^((#‘𝑊) + (#‘〈“𝑆”〉)))) |
13 | eleq1 2676 | . . . . . 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 246 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 = (#‘𝑊)) → 𝐼 ∈ ((#‘𝑊)..^((#‘𝑊) + (#‘〈“𝑆”〉)))) |
16 | 15 | 3adant2 1073 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → 𝐼 ∈ ((#‘𝑊)..^((#‘𝑊) + (#‘〈“𝑆”〉)))) |
17 | ccatval2 13215 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉 ∧ 𝐼 ∈ ((#‘𝑊)..^((#‘𝑊) + (#‘〈“𝑆”〉)))) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = (〈“𝑆”〉‘(𝐼 − (#‘𝑊)))) | |
18 | 1, 3, 16, 17 | syl3anc 1318 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = (〈“𝑆”〉‘(𝐼 − (#‘𝑊)))) |
19 | oveq1 6556 | . . . . 5 ⊢ (𝐼 = (#‘𝑊) → (𝐼 − (#‘𝑊)) = ((#‘𝑊) − (#‘𝑊))) | |
20 | 19 | 3ad2ant3 1077 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → (𝐼 − (#‘𝑊)) = ((#‘𝑊) − (#‘𝑊))) |
21 | 4 | nn0cnd 11230 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℂ) |
22 | 21 | subidd 10259 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((#‘𝑊) − (#‘𝑊)) = 0) |
23 | 22 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → ((#‘𝑊) − (#‘𝑊)) = 0) |
24 | 20, 23 | eqtrd 2644 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → (𝐼 − (#‘𝑊)) = 0) |
25 | 24 | fveq2d 6107 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → (〈“𝑆”〉‘(𝐼 − (#‘𝑊))) = (〈“𝑆”〉‘0)) |
26 | s1fv 13243 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉‘0) = 𝑆) | |
27 | 26 | 3ad2ant2 1076 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → (〈“𝑆”〉‘0) = 𝑆) |
28 | 18, 25, 27 | 3eqtrd 2648 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (#‘𝑊)) → ((𝑊 ++ 〈“𝑆”〉)‘𝐼) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℤcz 11254 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 〈“cs1 13149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 |
This theorem is referenced by: ccatws1ls 13262 ccatw2s1p1 13265 ccatw2s1p2 13266 gsmsymgrfixlem1 17670 gsmsymgreqlem2 17674 wwlknext 26252 wwlksnext 41099 |
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