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Mirrors > Home > ILE Home > Th. List > iccdili | GIF version |
Description: Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
iccdili.1 | ⊢ 𝐴 ∈ ℝ |
iccdili.2 | ⊢ 𝐵 ∈ ℝ |
iccdili.3 | ⊢ 𝑅 ∈ ℝ+ |
iccdili.4 | ⊢ (𝐴 · 𝑅) = 𝐶 |
iccdili.5 | ⊢ (𝐵 · 𝑅) = 𝐷 |
Ref | Expression |
---|---|
iccdili | ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 · 𝑅) ∈ (𝐶[,]𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccdili.1 | . . . 4 ⊢ 𝐴 ∈ ℝ | |
2 | iccdili.2 | . . . 4 ⊢ 𝐵 ∈ ℝ | |
3 | iccssre 9738 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 422 | . . 3 ⊢ (𝐴[,]𝐵) ⊆ ℝ |
5 | 4 | sseli 3093 | . 2 ⊢ (𝑋 ∈ (𝐴[,]𝐵) → 𝑋 ∈ ℝ) |
6 | iccdili.3 | . . . 4 ⊢ 𝑅 ∈ ℝ+ | |
7 | iccdili.4 | . . . . . 6 ⊢ (𝐴 · 𝑅) = 𝐶 | |
8 | iccdili.5 | . . . . . 6 ⊢ (𝐵 · 𝑅) = 𝐷 | |
9 | 7, 8 | iccdil 9781 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 · 𝑅) ∈ (𝐶[,]𝐷))) |
10 | 1, 2, 9 | mpanl12 432 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 · 𝑅) ∈ (𝐶[,]𝐷))) |
11 | 6, 10 | mpan2 421 | . . 3 ⊢ (𝑋 ∈ ℝ → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 · 𝑅) ∈ (𝐶[,]𝐷))) |
12 | 11 | biimpd 143 | . 2 ⊢ (𝑋 ∈ ℝ → (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 · 𝑅) ∈ (𝐶[,]𝐷))) |
13 | 5, 12 | mpcom 36 | 1 ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 · 𝑅) ∈ (𝐶[,]𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ⊆ wss 3071 (class class class)co 5774 ℝcr 7619 · cmul 7625 ℝ+crp 9441 [,]cicc 9674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-rp 9442 df-icc 9678 |
This theorem is referenced by: (None) |
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