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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 11317 | . . . . 5 ⊢ 𝐴 ∈ ℝ* |
3 | oprpiece1.2 | . . . . . 6 ⊢ 𝐵 ∈ ℝ | |
4 | 3 | rexri 11317 | . . . . 5 ⊢ 𝐵 ∈ ℝ* |
5 | oprpiece1.3 | . . . . 5 ⊢ 𝐴 ≤ 𝐵 | |
6 | lbicc2 13501 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
7 | 2, 4, 5, 6 | mp3an 1460 | . . . 4 ⊢ 𝐴 ∈ (𝐴[,]𝐵) |
8 | oprpiece1.6 | . . . 4 ⊢ 𝐾 ∈ (𝐴[,]𝐵) | |
9 | iccss2 13455 | . . . 4 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝐾 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵)) | |
10 | 7, 8, 9 | mp2an 692 | . . 3 ⊢ (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵) |
11 | ssid 4018 | . . 3 ⊢ 𝐶 ⊆ 𝐶 | |
12 | resmpo 7553 | . . 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 692 | . 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 5996 | . 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 13472 | . . . . . . . 8 ⊢ (𝐾 ∈ (𝐴[,]𝐵) → 𝐾 ∈ ℝ*) | |
18 | 8, 17 | ax-mp 5 | . . . . . . 7 ⊢ 𝐾 ∈ ℝ* |
19 | iccleub 13439 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐾 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐾)) → 𝑥 ≤ 𝐾) | |
20 | 2, 18, 19 | mp3an12 1450 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴[,]𝐾) → 𝑥 ≤ 𝐾) |
21 | 20 | iftrued 4539 | . . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐾) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑅) |
22 | 21 | adantr 480 | . . . 4 ⊢ ((𝑥 ∈ (𝐴[,]𝐾) ∧ 𝑦 ∈ 𝐶) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑅) |
23 | 22 | mpoeq3ia 7511 | . . 3 ⊢ (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ 𝑅) |
24 | 16, 23 | eqtr4i 2766 | . 2 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) |
25 | 13, 15, 24 | 3eqtr4i 2773 | 1 ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ifcif 4531 class class class wbr 5148 × cxp 5687 ↾ cres 5691 (class class class)co 7431 ∈ cmpo 7433 ℝcr 11152 ℝ*cxr 11292 ≤ cle 11294 [,]cicc 13387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-icc 13391 |
This theorem is referenced by: (None) |
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