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Mirrors > Home > MPE Home > Th. List > ccatlid | Structured version Visualization version GIF version |
Description: Concatenation of a word by the empty word on the left. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.) |
Ref | Expression |
---|---|
ccatlid | ⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrd0 14529 | . . . 4 ⊢ ∅ ∈ Word 𝐵 | |
2 | ccatvalfn 14571 | . . . 4 ⊢ ((∅ ∈ Word 𝐵 ∧ 𝑆 ∈ Word 𝐵) → (∅ ++ 𝑆) Fn (0..^((♯‘∅) + (♯‘𝑆)))) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) Fn (0..^((♯‘∅) + (♯‘𝑆)))) |
4 | hash0 14366 | . . . . . . . 8 ⊢ (♯‘∅) = 0 | |
5 | 4 | oveq1i 7436 | . . . . . . 7 ⊢ ((♯‘∅) + (♯‘𝑆)) = (0 + (♯‘𝑆)) |
6 | lencl 14523 | . . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) ∈ ℕ0) | |
7 | 6 | nn0cnd 12572 | . . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) ∈ ℂ) |
8 | 7 | addlidd 11453 | . . . . . . 7 ⊢ (𝑆 ∈ Word 𝐵 → (0 + (♯‘𝑆)) = (♯‘𝑆)) |
9 | 5, 8 | eqtrid 2780 | . . . . . 6 ⊢ (𝑆 ∈ Word 𝐵 → ((♯‘∅) + (♯‘𝑆)) = (♯‘𝑆)) |
10 | 9 | eqcomd 2734 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) = ((♯‘∅) + (♯‘𝑆))) |
11 | 10 | oveq2d 7442 | . . . 4 ⊢ (𝑆 ∈ Word 𝐵 → (0..^(♯‘𝑆)) = (0..^((♯‘∅) + (♯‘𝑆)))) |
12 | 11 | fneq2d 6653 | . . 3 ⊢ (𝑆 ∈ Word 𝐵 → ((∅ ++ 𝑆) Fn (0..^(♯‘𝑆)) ↔ (∅ ++ 𝑆) Fn (0..^((♯‘∅) + (♯‘𝑆))))) |
13 | 3, 12 | mpbird 256 | . 2 ⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) Fn (0..^(♯‘𝑆))) |
14 | wrdfn 14518 | . 2 ⊢ (𝑆 ∈ Word 𝐵 → 𝑆 Fn (0..^(♯‘𝑆))) | |
15 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘∅) = 0) |
16 | 15, 9 | oveq12d 7444 | . . . . . 6 ⊢ (𝑆 ∈ Word 𝐵 → ((♯‘∅)..^((♯‘∅) + (♯‘𝑆))) = (0..^(♯‘𝑆))) |
17 | 16 | eleq2d 2815 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐵 → (𝑥 ∈ ((♯‘∅)..^((♯‘∅) + (♯‘𝑆))) ↔ 𝑥 ∈ (0..^(♯‘𝑆)))) |
18 | 17 | biimpar 476 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ((♯‘∅)..^((♯‘∅) + (♯‘𝑆)))) |
19 | ccatval2 14568 | . . . . 5 ⊢ ((∅ ∈ Word 𝐵 ∧ 𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ ((♯‘∅)..^((♯‘∅) + (♯‘𝑆)))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘(𝑥 − (♯‘∅)))) | |
20 | 1, 19 | mp3an1 1444 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ ((♯‘∅)..^((♯‘∅) + (♯‘𝑆)))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘(𝑥 − (♯‘∅)))) |
21 | 18, 20 | syldan 589 | . . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘(𝑥 − (♯‘∅)))) |
22 | 4 | oveq2i 7437 | . . . . 5 ⊢ (𝑥 − (♯‘∅)) = (𝑥 − 0) |
23 | elfzoelz 13672 | . . . . . . . 8 ⊢ (𝑥 ∈ (0..^(♯‘𝑆)) → 𝑥 ∈ ℤ) | |
24 | 23 | adantl 480 | . . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ℤ) |
25 | 24 | zcnd 12705 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ℂ) |
26 | 25 | subid1d 11598 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑥 − 0) = 𝑥) |
27 | 22, 26 | eqtrid 2780 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑥 − (♯‘∅)) = 𝑥) |
28 | 27 | fveq2d 6906 | . . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑆‘(𝑥 − (♯‘∅))) = (𝑆‘𝑥)) |
29 | 21, 28 | eqtrd 2768 | . 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘𝑥)) |
30 | 13, 14, 29 | eqfnfvd 7048 | 1 ⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∅c0 4326 Fn wfn 6548 ‘cfv 6553 (class class class)co 7426 0cc0 11146 + caddc 11149 − cmin 11482 ℤcz 12596 ..^cfzo 13667 ♯chash 14329 Word cword 14504 ++ cconcat 14560 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-concat 14561 |
This theorem is referenced by: ccatidid 14580 ccat1st1st 14618 swrdccat 14725 s0s1 14913 gsumccat 18800 frmdmnd 18818 frmd0 18819 efgcpbl2 19719 frgp0 19722 frgpnabllem1 19835 signstfvneq0 34237 lpadlen1 34344 |
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