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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 10097 | . . . . 5 ⊢ 𝐴 ∈ ℝ* |
3 | oprpiece1.2 | . . . . . 6 ⊢ 𝐵 ∈ ℝ | |
4 | 3 | rexri 10097 | . . . . 5 ⊢ 𝐵 ∈ ℝ* |
5 | oprpiece1.3 | . . . . 5 ⊢ 𝐴 ≤ 𝐵 | |
6 | lbicc2 12288 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
7 | 2, 4, 5, 6 | mp3an 1424 | . . . 4 ⊢ 𝐴 ∈ (𝐴[,]𝐵) |
8 | oprpiece1.6 | . . . 4 ⊢ 𝐾 ∈ (𝐴[,]𝐵) | |
9 | iccss2 12244 | . . . 4 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝐾 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵)) | |
10 | 7, 8, 9 | mp2an 708 | . . 3 ⊢ (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵) |
11 | ssid 3624 | . . 3 ⊢ 𝐶 ⊆ 𝐶 | |
12 | resmpt2 6758 | . . 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 708 | . 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 5392 | . 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 12219 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐾 ∈ (𝐴[,]𝐵) ↔ (𝐾 ∈ ℝ* ∧ 𝐴 ≤ 𝐾 ∧ 𝐾 ≤ 𝐵))) | |
18 | 2, 4, 17 | mp2an 708 | . . . . . . . . 9 ⊢ (𝐾 ∈ (𝐴[,]𝐵) ↔ (𝐾 ∈ ℝ* ∧ 𝐴 ≤ 𝐾 ∧ 𝐾 ≤ 𝐵)) |
19 | 18 | simp1bi 1076 | . . . . . . . 8 ⊢ (𝐾 ∈ (𝐴[,]𝐵) → 𝐾 ∈ ℝ*) |
20 | 8, 19 | ax-mp 5 | . . . . . . 7 ⊢ 𝐾 ∈ ℝ* |
21 | iccleub 12229 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐾 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐾)) → 𝑥 ≤ 𝐾) | |
22 | 2, 20, 21 | mp3an12 1414 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴[,]𝐾) → 𝑥 ≤ 𝐾) |
23 | 22 | iftrued 4094 | . . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐾) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑅) |
24 | 23 | adantr 481 | . . . 4 ⊢ ((𝑥 ∈ (𝐴[,]𝐾) ∧ 𝑦 ∈ 𝐶) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑅) |
25 | 24 | mpt2eq3ia 6720 | . . 3 ⊢ (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ 𝑅) |
26 | 16, 25 | eqtr4i 2647 | . 2 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) |
27 | 13, 15, 26 | 3eqtr4i 2654 | 1 ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ifcif 4086 class class class wbr 4653 × cxp 5112 ↾ cres 5116 (class class class)co 6650 ↦ cmpt2 6652 ℝcr 9935 ℝ*cxr 10073 ≤ cle 10075 [,]cicc 12178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-icc 12182 |
This theorem is referenced by: (None) |
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