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Mirrors > Home > MPE Home > Th. List > ccatfval | Structured version Visualization version GIF version |
Description: Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Ref | Expression |
---|---|
ccatfval | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3459 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
2 | elex 3459 | . 2 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V) | |
3 | fveq2 6645 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (♯‘𝑠) = (♯‘𝑆)) | |
4 | fveq2 6645 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (♯‘𝑡) = (♯‘𝑇)) | |
5 | 3, 4 | oveqan12d 7154 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((♯‘𝑠) + (♯‘𝑡)) = ((♯‘𝑆) + (♯‘𝑇))) |
6 | 5 | oveq2d 7151 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (0..^((♯‘𝑠) + (♯‘𝑡))) = (0..^((♯‘𝑆) + (♯‘𝑇)))) |
7 | 3 | oveq2d 7151 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (0..^(♯‘𝑠)) = (0..^(♯‘𝑆))) |
8 | 7 | eleq2d 2875 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ (0..^(♯‘𝑠)) ↔ 𝑥 ∈ (0..^(♯‘𝑆)))) |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑥 ∈ (0..^(♯‘𝑠)) ↔ 𝑥 ∈ (0..^(♯‘𝑆)))) |
10 | fveq1 6644 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑥) = (𝑆‘𝑥)) | |
11 | 10 | adantr 484 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠‘𝑥) = (𝑆‘𝑥)) |
12 | simpr 488 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → 𝑡 = 𝑇) | |
13 | 3 | oveq2d 7151 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑥 − (♯‘𝑠)) = (𝑥 − (♯‘𝑆))) |
14 | 13 | adantr 484 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑥 − (♯‘𝑠)) = (𝑥 − (♯‘𝑆))) |
15 | 12, 14 | fveq12d 6652 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑡‘(𝑥 − (♯‘𝑠))) = (𝑇‘(𝑥 − (♯‘𝑆)))) |
16 | 9, 11, 15 | ifbieq12d 4452 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠)))) = if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))) |
17 | 6, 16 | mpteq12dv 5115 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))))) |
18 | df-concat 13914 | . . 3 ⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠)))))) | |
19 | ovex 7168 | . . . 4 ⊢ (0..^((♯‘𝑆) + (♯‘𝑇))) ∈ V | |
20 | 19 | mptex 6963 | . . 3 ⊢ (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))) ∈ V |
21 | 17, 18, 20 | ovmpoa 7284 | . 2 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))))) |
22 | 1, 2, 21 | syl2an 598 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ifcif 4425 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 0cc0 10526 + caddc 10529 − cmin 10859 ..^cfzo 13028 ♯chash 13686 ++ cconcat 13913 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-concat 13914 |
This theorem is referenced by: ccatcl 13917 ccatlen 13918 ccatlenOLD 13919 ccatval1 13921 ccatval1OLD 13922 ccatval2 13923 ccatvalfn 13926 ccatalpha 13938 repswccat 14139 ccatco 14188 ofccat 14320 ccatmulgnn0dir 31922 |
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