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Mirrors > Home > MPE Home > Th. List > ccatval1 | Structured version Visualization version GIF version |
Description: Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 30-Apr-2020.) (Revised by JJ, 18-Jan-2024.) |
Ref | Expression |
---|---|
ccatval1 | ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆‘𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccatfval 14525 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))))) | |
2 | 1 | 3adant3 1132 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))))) |
3 | eleq1 2821 | . . . 4 ⊢ (𝑥 = 𝐼 → (𝑥 ∈ (0..^(♯‘𝑆)) ↔ 𝐼 ∈ (0..^(♯‘𝑆)))) | |
4 | fveq2 6891 | . . . 4 ⊢ (𝑥 = 𝐼 → (𝑆‘𝑥) = (𝑆‘𝐼)) | |
5 | fvoveq1 7434 | . . . 4 ⊢ (𝑥 = 𝐼 → (𝑇‘(𝑥 − (♯‘𝑆))) = (𝑇‘(𝐼 − (♯‘𝑆)))) | |
6 | 3, 4, 5 | ifbieq12d 4556 | . . 3 ⊢ (𝑥 = 𝐼 → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))) = if(𝐼 ∈ (0..^(♯‘𝑆)), (𝑆‘𝐼), (𝑇‘(𝐼 − (♯‘𝑆))))) |
7 | iftrue 4534 | . . . 4 ⊢ (𝐼 ∈ (0..^(♯‘𝑆)) → if(𝐼 ∈ (0..^(♯‘𝑆)), (𝑆‘𝐼), (𝑇‘(𝐼 − (♯‘𝑆)))) = (𝑆‘𝐼)) | |
8 | 7 | 3ad2ant3 1135 | . . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → if(𝐼 ∈ (0..^(♯‘𝑆)), (𝑆‘𝐼), (𝑇‘(𝐼 − (♯‘𝑆)))) = (𝑆‘𝐼)) |
9 | 6, 8 | sylan9eqr 2794 | . 2 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) ∧ 𝑥 = 𝐼) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))) = (𝑆‘𝐼)) |
10 | id 22 | . . . 4 ⊢ (𝐼 ∈ (0..^(♯‘𝑆)) → 𝐼 ∈ (0..^(♯‘𝑆))) | |
11 | lencl 14485 | . . . 4 ⊢ (𝑇 ∈ Word 𝐵 → (♯‘𝑇) ∈ ℕ0) | |
12 | elfzoext 13691 | . . . 4 ⊢ ((𝐼 ∈ (0..^(♯‘𝑆)) ∧ (♯‘𝑇) ∈ ℕ0) → 𝐼 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) | |
13 | 10, 11, 12 | syl2anr 597 | . . 3 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → 𝐼 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) |
14 | 13 | 3adant1 1130 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → 𝐼 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) |
15 | fvexd 6906 | . 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → (𝑆‘𝐼) ∈ V) | |
16 | 2, 9, 14, 15 | fvmptd 7005 | 1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ifcif 4528 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7411 0cc0 11112 + caddc 11115 − cmin 11446 ℕ0cn0 12474 ..^cfzo 13629 ♯chash 14292 Word cword 14466 ++ cconcat 14522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-fzo 13630 df-hash 14293 df-word 14467 df-concat 14523 |
This theorem is referenced by: ccatsymb 14534 ccatfv0 14535 ccatval1lsw 14536 ccatrid 14539 ccatass 14540 ccatrn 14541 ccats1val1 14578 ccat2s1p1 14581 lswccats1fst 14587 ccat2s1fvw 14590 ccatswrd 14620 ccatpfx 14653 pfxccat1 14654 swrdccatin1 14677 pfxccatin12lem3 14684 pfxccatin12 14685 splfv1 14707 splfv2a 14708 revccat 14718 cshwidxmod 14755 cats1fv 14812 ccat2s1fvwALT 14908 gsumsgrpccat 18723 efgsp1 19607 efgredlemd 19614 efgrelexlemb 19620 tgcgr4 27820 clwwlkccatlem 29280 clwwlkel 29337 wwlksext2clwwlk 29348 ccatf1 32153 cycpmco2lem2 32327 cycpmco2lem4 32329 cycpmco2lem5 32330 cycpmco2 32333 signstfvn 33649 signstfvp 33651 signstfvneq0 33652 lpadleft 33764 |
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