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Mirrors > Home > MPE Home > Th. List > oprpiece1res1 | Structured version Visualization version GIF version |
Description: Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
oprpiece1.1 | ⊢ 𝐴 ∈ ℝ |
oprpiece1.2 | ⊢ 𝐵 ∈ ℝ |
oprpiece1.3 | ⊢ 𝐴 ≤ 𝐵 |
oprpiece1.4 | ⊢ 𝑅 ∈ V |
oprpiece1.5 | ⊢ 𝑆 ∈ V |
oprpiece1.6 | ⊢ 𝐾 ∈ (𝐴[,]𝐵) |
oprpiece1.7 | ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) |
oprpiece1.8 | ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ 𝑅) |
Ref | Expression |
---|---|
oprpiece1res1 | ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oprpiece1.1 | . . . . . 6 ⊢ 𝐴 ∈ ℝ | |
2 | 1 | rexri 10701 | . . . . 5 ⊢ 𝐴 ∈ ℝ* |
3 | oprpiece1.2 | . . . . . 6 ⊢ 𝐵 ∈ ℝ | |
4 | 3 | rexri 10701 | . . . . 5 ⊢ 𝐵 ∈ ℝ* |
5 | oprpiece1.3 | . . . . 5 ⊢ 𝐴 ≤ 𝐵 | |
6 | lbicc2 12855 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
7 | 2, 4, 5, 6 | mp3an 1457 | . . . 4 ⊢ 𝐴 ∈ (𝐴[,]𝐵) |
8 | oprpiece1.6 | . . . 4 ⊢ 𝐾 ∈ (𝐴[,]𝐵) | |
9 | iccss2 12810 | . . . 4 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝐾 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵)) | |
10 | 7, 8, 9 | mp2an 690 | . . 3 ⊢ (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵) |
11 | ssid 3991 | . . 3 ⊢ 𝐶 ⊆ 𝐶 | |
12 | resmpo 7274 | . . 3 ⊢ (((𝐴[,]𝐾) ⊆ (𝐴[,]𝐵) ∧ 𝐶 ⊆ 𝐶) → ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) ↾ ((𝐴[,]𝐾) × 𝐶)) = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))) | |
13 | 10, 11, 12 | mp2an 690 | . 2 ⊢ ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) ↾ ((𝐴[,]𝐾) × 𝐶)) = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) |
14 | oprpiece1.7 | . . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) | |
15 | 14 | reseq1i 5851 | . 2 ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) ↾ ((𝐴[,]𝐾) × 𝐶)) |
16 | oprpiece1.8 | . . 3 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ 𝑅) | |
17 | eliccxr 12826 | . . . . . . . 8 ⊢ (𝐾 ∈ (𝐴[,]𝐵) → 𝐾 ∈ ℝ*) | |
18 | 8, 17 | ax-mp 5 | . . . . . . 7 ⊢ 𝐾 ∈ ℝ* |
19 | iccleub 12795 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐾 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐾)) → 𝑥 ≤ 𝐾) | |
20 | 2, 18, 19 | mp3an12 1447 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴[,]𝐾) → 𝑥 ≤ 𝐾) |
21 | 20 | iftrued 4477 | . . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐾) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑅) |
22 | 21 | adantr 483 | . . . 4 ⊢ ((𝑥 ∈ (𝐴[,]𝐾) ∧ 𝑦 ∈ 𝐶) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑅) |
23 | 22 | mpoeq3ia 7234 | . . 3 ⊢ (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ 𝑅) |
24 | 16, 23 | eqtr4i 2849 | . 2 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) |
25 | 13, 15, 24 | 3eqtr4i 2856 | 1 ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ifcif 4469 class class class wbr 5068 × cxp 5555 ↾ cres 5559 (class class class)co 7158 ∈ cmpo 7160 ℝcr 10538 ℝ*cxr 10676 ≤ cle 10678 [,]cicc 12744 |
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-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
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-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 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-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-icc 12748 |
This theorem is referenced by: (None) |
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