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Mirrors > Home > MPE Home > Th. List > iccshftri | Structured version Visualization version GIF version |
Description: Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
iccshftri.1 | ⊢ 𝐴 ∈ ℝ |
iccshftri.2 | ⊢ 𝐵 ∈ ℝ |
iccshftri.3 | ⊢ 𝑅 ∈ ℝ |
iccshftri.4 | ⊢ (𝐴 + 𝑅) = 𝐶 |
iccshftri.5 | ⊢ (𝐵 + 𝑅) = 𝐷 |
Ref | Expression |
---|---|
iccshftri | ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 + 𝑅) ∈ (𝐶[,]𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccshftri.1 | . . . 4 ⊢ 𝐴 ∈ ℝ | |
2 | iccshftri.2 | . . . 4 ⊢ 𝐵 ∈ ℝ | |
3 | iccssre 12542 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 685 | . . 3 ⊢ (𝐴[,]𝐵) ⊆ ℝ |
5 | 4 | sseli 3822 | . 2 ⊢ (𝑋 ∈ (𝐴[,]𝐵) → 𝑋 ∈ ℝ) |
6 | iccshftri.3 | . . . 4 ⊢ 𝑅 ∈ ℝ | |
7 | iccshftri.4 | . . . . . 6 ⊢ (𝐴 + 𝑅) = 𝐶 | |
8 | iccshftri.5 | . . . . . 6 ⊢ (𝐵 + 𝑅) = 𝐷 | |
9 | 7, 8 | iccshftr 12598 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))) |
10 | 1, 2, 9 | mpanl12 695 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))) |
11 | 6, 10 | mpan2 684 | . . 3 ⊢ (𝑋 ∈ ℝ → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))) |
12 | 11 | biimpd 221 | . 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 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ⊆ wss 3797 (class class class)co 6904 ℝcr 10250 + caddc 10254 [,]cicc 12465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-icc 12469 |
This theorem is referenced by: pcoass 23192 |
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