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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 9976 | . . . . 5 ⊢ 𝐴 ∈ ℝ* |
3 | oprpiece1.2 | . . . . . 6 ⊢ 𝐵 ∈ ℝ | |
4 | 3 | rexri 9976 | . . . . 5 ⊢ 𝐵 ∈ ℝ* |
5 | oprpiece1.3 | . . . . 5 ⊢ 𝐴 ≤ 𝐵 | |
6 | lbicc2 12159 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
7 | 2, 4, 5, 6 | mp3an 1416 | . . . 4 ⊢ 𝐴 ∈ (𝐴[,]𝐵) |
8 | oprpiece1.6 | . . . 4 ⊢ 𝐾 ∈ (𝐴[,]𝐵) | |
9 | iccss2 12115 | . . . 4 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝐾 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵)) | |
10 | 7, 8, 9 | mp2an 704 | . . 3 ⊢ (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵) |
11 | ssid 3587 | . . 3 ⊢ 𝐶 ⊆ 𝐶 | |
12 | resmpt2 6656 | . . 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 704 | . 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 5313 | . 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 | elicc1 12090 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐾 ∈ (𝐴[,]𝐵) ↔ (𝐾 ∈ ℝ* ∧ 𝐴 ≤ 𝐾 ∧ 𝐾 ≤ 𝐵))) | |
18 | 2, 4, 17 | mp2an 704 | . . . . . . . . 9 ⊢ (𝐾 ∈ (𝐴[,]𝐵) ↔ (𝐾 ∈ ℝ* ∧ 𝐴 ≤ 𝐾 ∧ 𝐾 ≤ 𝐵)) |
19 | 18 | simp1bi 1069 | . . . . . . . 8 ⊢ (𝐾 ∈ (𝐴[,]𝐵) → 𝐾 ∈ ℝ*) |
20 | 8, 19 | ax-mp 5 | . . . . . . 7 ⊢ 𝐾 ∈ ℝ* |
21 | iccleub 12100 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐾 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐾)) → 𝑥 ≤ 𝐾) | |
22 | 2, 20, 21 | mp3an12 1406 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴[,]𝐾) → 𝑥 ≤ 𝐾) |
23 | 22 | iftrued 4044 | . . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐾) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑅) |
24 | 23 | adantr 480 | . . . 4 ⊢ ((𝑥 ∈ (𝐴[,]𝐾) ∧ 𝑦 ∈ 𝐶) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑅) |
25 | 24 | mpt2eq3ia 6618 | . . 3 ⊢ (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ 𝑅) |
26 | 16, 25 | eqtr4i 2635 | . 2 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) |
27 | 13, 15, 26 | 3eqtr4i 2642 | 1 ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ifcif 4036 class class class wbr 4583 × cxp 5036 ↾ cres 5040 (class class class)co 6549 ↦ cmpt2 6551 ℝcr 9814 ℝ*cxr 9952 ≤ cle 9954 [,]cicc 12049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-icc 12053 |
This theorem is referenced by: (None) |
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