![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > icccntri | Structured version Visualization version GIF version |
Description: Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
icccntri.1 | ⊢ 𝐴 ∈ ℝ |
icccntri.2 | ⊢ 𝐵 ∈ ℝ |
icccntri.3 | ⊢ 𝑅 ∈ ℝ+ |
icccntri.4 | ⊢ (𝐴 / 𝑅) = 𝐶 |
icccntri.5 | ⊢ (𝐵 / 𝑅) = 𝐷 |
Ref | Expression |
---|---|
icccntri | ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 / 𝑅) ∈ (𝐶[,]𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icccntri.1 | . . . 4 ⊢ 𝐴 ∈ ℝ | |
2 | icccntri.2 | . . . 4 ⊢ 𝐵 ∈ ℝ | |
3 | iccssre 13465 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 692 | . . 3 ⊢ (𝐴[,]𝐵) ⊆ ℝ |
5 | 4 | sseli 3990 | . 2 ⊢ (𝑋 ∈ (𝐴[,]𝐵) → 𝑋 ∈ ℝ) |
6 | icccntri.3 | . . . 4 ⊢ 𝑅 ∈ ℝ+ | |
7 | icccntri.4 | . . . . . 6 ⊢ (𝐴 / 𝑅) = 𝐶 | |
8 | icccntri.5 | . . . . . 6 ⊢ (𝐵 / 𝑅) = 𝐷 | |
9 | 7, 8 | icccntr 13528 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))) |
10 | 1, 2, 9 | mpanl12 702 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))) |
11 | 6, 10 | mpan2 691 | . . 3 ⊢ (𝑋 ∈ ℝ → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))) |
12 | 11 | biimpd 229 | . 2 ⊢ (𝑋 ∈ ℝ → (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))) |
13 | 5, 12 | mpcom 38 | 1 ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 / 𝑅) ∈ (𝐶[,]𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 (class class class)co 7430 ℝcr 11151 / cdiv 11917 ℝ+crp 13031 [,]cicc 13386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-rp 13032 df-icc 13390 |
This theorem is referenced by: pcoass 25070 |
Copyright terms: Public domain | W3C validator |