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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 13891 | . . . 4 ⊢ ∅ ∈ Word 𝐵 | |
2 | ccatvalfn 13937 | . . . 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 13731 | . . . . . . . 8 ⊢ (♯‘∅) = 0 | |
5 | 4 | oveq1i 7168 | . . . . . . 7 ⊢ ((♯‘∅) + (♯‘𝑆)) = (0 + (♯‘𝑆)) |
6 | lencl 13885 | . . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) ∈ ℕ0) | |
7 | 6 | nn0cnd 11960 | . . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) ∈ ℂ) |
8 | 7 | addid2d 10843 | . . . . . . 7 ⊢ (𝑆 ∈ Word 𝐵 → (0 + (♯‘𝑆)) = (♯‘𝑆)) |
9 | 5, 8 | syl5eq 2870 | . . . . . 6 ⊢ (𝑆 ∈ Word 𝐵 → ((♯‘∅) + (♯‘𝑆)) = (♯‘𝑆)) |
10 | 9 | eqcomd 2829 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) = ((♯‘∅) + (♯‘𝑆))) |
11 | 10 | oveq2d 7174 | . . . 4 ⊢ (𝑆 ∈ Word 𝐵 → (0..^(♯‘𝑆)) = (0..^((♯‘∅) + (♯‘𝑆)))) |
12 | 11 | fneq2d 6449 | . . 3 ⊢ (𝑆 ∈ Word 𝐵 → ((∅ ++ 𝑆) Fn (0..^(♯‘𝑆)) ↔ (∅ ++ 𝑆) Fn (0..^((♯‘∅) + (♯‘𝑆))))) |
13 | 3, 12 | mpbird 259 | . 2 ⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) Fn (0..^(♯‘𝑆))) |
14 | wrdfn 13879 | . 2 ⊢ (𝑆 ∈ Word 𝐵 → 𝑆 Fn (0..^(♯‘𝑆))) | |
15 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘∅) = 0) |
16 | 15, 9 | oveq12d 7176 | . . . . . 6 ⊢ (𝑆 ∈ Word 𝐵 → ((♯‘∅)..^((♯‘∅) + (♯‘𝑆))) = (0..^(♯‘𝑆))) |
17 | 16 | eleq2d 2900 | . . . . 5 ⊢ (𝑆 ∈ Word 𝐵 → (𝑥 ∈ ((♯‘∅)..^((♯‘∅) + (♯‘𝑆))) ↔ 𝑥 ∈ (0..^(♯‘𝑆)))) |
18 | 17 | biimpar 480 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ((♯‘∅)..^((♯‘∅) + (♯‘𝑆)))) |
19 | ccatval2 13934 | . . . . 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 593 | . . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘(𝑥 − (♯‘∅)))) |
22 | 4 | oveq2i 7169 | . . . . 5 ⊢ (𝑥 − (♯‘∅)) = (𝑥 − 0) |
23 | elfzoelz 13041 | . . . . . . . 8 ⊢ (𝑥 ∈ (0..^(♯‘𝑆)) → 𝑥 ∈ ℤ) | |
24 | 23 | adantl 484 | . . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ℤ) |
25 | 24 | zcnd 12091 | . . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ℂ) |
26 | 25 | subid1d 10988 | . . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑥 − 0) = 𝑥) |
27 | 22, 26 | syl5eq 2870 | . . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑥 − (♯‘∅)) = 𝑥) |
28 | 27 | fveq2d 6676 | . . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑆‘(𝑥 − (♯‘∅))) = (𝑆‘𝑥)) |
29 | 21, 28 | eqtrd 2858 | . 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((∅ ++ 𝑆)‘𝑥) = (𝑆‘𝑥)) |
30 | 13, 14, 29 | eqfnfvd 6807 | 1 ⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∅c0 4293 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 0cc0 10539 + caddc 10542 − cmin 10872 ℤcz 11984 ..^cfzo 13036 ♯chash 13693 Word cword 13864 ++ cconcat 13924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 |
This theorem is referenced by: ccatidid 13946 ccat1st1st 13986 swrdccat 14099 s0s1 14286 gsumccatOLD 18007 gsumccat 18008 frmdmnd 18026 frmd0 18027 efgcpbl2 18885 frgp0 18888 frgpnabllem1 18995 signstfvneq0 31844 lpadlen1 31952 |
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