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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 13362 | . . . 4 ⊢ ∅ ∈ Word 𝐵 | |
2 | ccatvalfn 13399 | . . . 4 ⊢ ((∅ ∈ Word 𝐵 ∧ 𝑆 ∈ Word 𝐵) → (∅ ++ 𝑆) Fn (0..^((#‘∅) + (#‘𝑆)))) | |
3 | 1, 2 | mpan 706 | . . 3 ⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) Fn (0..^((#‘∅) + (#‘𝑆)))) |
4 | hash0 13196 | . . . . . . . 8 ⊢ (#‘∅) = 0 | |
5 | 4 | oveq1i 6700 | . . . . . . 7 ⊢ ((#‘∅) + (#‘𝑆)) = (0 + (#‘𝑆)) |
6 | lencl 13356 | . . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐵 → (#‘𝑆) ∈ ℕ0) | |
7 | 6 | nn0cnd 11391 | . . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐵 → (#‘𝑆) ∈ ℂ) |
8 | 7 | addid2d 10275 | . . . . . . 7 ⊢ (𝑆 ∈ Word 𝐵 → (0 + (#‘𝑆)) = (#‘𝑆)) |
9 | 5, 8 | syl5eq 2697 | . . . . . 6 ⊢ (𝑆 ∈ Word 𝐵 → ((#‘∅) + (#‘𝑆)) = (#‘𝑆)) |
10 | 9 | eqcomd 2657 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐵 → (#‘𝑆) = ((#‘∅) + (#‘𝑆))) |
11 | 10 | oveq2d 6706 | . . . 4 ⊢ (𝑆 ∈ Word 𝐵 → (0..^(#‘𝑆)) = (0..^((#‘∅) + (#‘𝑆)))) |
12 | 11 | fneq2d 6020 | . . 3 ⊢ (𝑆 ∈ Word 𝐵 → ((∅ ++ 𝑆) Fn (0..^(#‘𝑆)) ↔ (∅ ++ 𝑆) Fn (0..^((#‘∅) + (#‘𝑆))))) |
13 | 3, 12 | mpbird 247 | . 2 ⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) Fn (0..^(#‘𝑆))) |
14 | wrdfn 13351 | . 2 ⊢ (𝑆 ∈ Word 𝐵 → 𝑆 Fn (0..^(#‘𝑆))) | |
15 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝑆 ∈ Word 𝐵 → (#‘∅) = 0) |
16 | 15, 9 | oveq12d 6708 | . . . . . 6 ⊢ (𝑆 ∈ Word 𝐵 → ((#‘∅)..^((#‘∅) + (#‘𝑆))) = (0..^(#‘𝑆))) |
17 | 16 | eleq2d 2716 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐵 → (𝑥 ∈ ((#‘∅)..^((#‘∅) + (#‘𝑆))) ↔ 𝑥 ∈ (0..^(#‘𝑆)))) |
18 | 17 | biimpar 501 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑥 ∈ ((#‘∅)..^((#‘∅) + (#‘𝑆)))) |
19 | ccatval2 13396 | . . . . 5 ⊢ ((∅ ∈ Word 𝐵 ∧ 𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ ((#‘∅)..^((#‘∅) + (#‘𝑆)))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘(𝑥 − (#‘∅)))) | |
20 | 1, 19 | mp3an1 1451 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ ((#‘∅)..^((#‘∅) + (#‘𝑆)))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘(𝑥 − (#‘∅)))) |
21 | 18, 20 | syldan 486 | . . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘(𝑥 − (#‘∅)))) |
22 | 4 | oveq2i 6701 | . . . . 5 ⊢ (𝑥 − (#‘∅)) = (𝑥 − 0) |
23 | elfzoelz 12509 | . . . . . . . 8 ⊢ (𝑥 ∈ (0..^(#‘𝑆)) → 𝑥 ∈ ℤ) | |
24 | 23 | adantl 481 | . . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑥 ∈ ℤ) |
25 | 24 | zcnd 11521 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑥 ∈ ℂ) |
26 | 25 | subid1d 10419 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → (𝑥 − 0) = 𝑥) |
27 | 22, 26 | syl5eq 2697 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → (𝑥 − (#‘∅)) = 𝑥) |
28 | 27 | fveq2d 6233 | . . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → (𝑆‘(𝑥 − (#‘∅))) = (𝑆‘𝑥)) |
29 | 21, 28 | eqtrd 2685 | . 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘𝑥)) |
30 | 13, 14, 29 | eqfnfvd 6354 | 1 ⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∅c0 3948 Fn wfn 5921 ‘cfv 5926 (class class class)co 6690 0cc0 9974 + caddc 9977 − cmin 10304 ℤcz 11415 ..^cfzo 12504 #chash 13157 Word cword 13323 ++ cconcat 13325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-concat 13333 |
This theorem is referenced by: ccatidid 13408 ccat1st1st 13448 swrdccat 13539 swrdccat3a 13540 s0s1 13713 gsumccat 17425 frmdmnd 17443 frmd0 17444 efginvrel2 18186 efgcpbl2 18216 frgp0 18219 frgpnabllem1 18322 signstfvneq0 30777 elmrsubrn 31543 |
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