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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 11319 | . . . . 5 ⊢ 𝐴 ∈ ℝ* |
| 3 | oprpiece1.2 | . . . . . 6 ⊢ 𝐵 ∈ ℝ | |
| 4 | 3 | rexri 11319 | . . . . 5 ⊢ 𝐵 ∈ ℝ* |
| 5 | oprpiece1.3 | . . . . 5 ⊢ 𝐴 ≤ 𝐵 | |
| 6 | lbicc2 13504 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
| 7 | 2, 4, 5, 6 | mp3an 1463 | . . . 4 ⊢ 𝐴 ∈ (𝐴[,]𝐵) |
| 8 | oprpiece1.6 | . . . 4 ⊢ 𝐾 ∈ (𝐴[,]𝐵) | |
| 9 | iccss2 13458 | . . . 4 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝐾 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵)) | |
| 10 | 7, 8, 9 | mp2an 692 | . . 3 ⊢ (𝐴[,]𝐾) ⊆ (𝐴[,]𝐵) |
| 11 | ssid 4006 | . . 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 5993 | . 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 13475 | . . . . . . . 8 ⊢ (𝐾 ∈ (𝐴[,]𝐵) → 𝐾 ∈ ℝ*) | |
| 18 | 8, 17 | ax-mp 5 | . . . . . . 7 ⊢ 𝐾 ∈ ℝ* |
| 19 | iccleub 13442 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐾 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐾)) → 𝑥 ≤ 𝐾) | |
| 20 | 2, 18, 19 | mp3an12 1453 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴[,]𝐾) → 𝑥 ≤ 𝐾) |
| 21 | 20 | iftrued 4533 | . . . . 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 2768 | . 2 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) |
| 25 | 13, 15, 24 | 3eqtr4i 2775 | 1 ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ifcif 4525 class class class wbr 5143 × cxp 5683 ↾ cres 5687 (class class class)co 7431 ∈ cmpo 7433 ℝcr 11154 ℝ*cxr 11294 ≤ cle 11296 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-icc 13394 |
| This theorem is referenced by: (None) |
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