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Mirrors > Home > ILE Home > Th. List > iccshftri | 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 9882 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 423 | . . 3 ⊢ (𝐴[,]𝐵) ⊆ ℝ |
5 | 4 | sseli 3133 | . 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 9921 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))) |
10 | 1, 2, 9 | mpanl12 433 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))) |
11 | 6, 10 | mpan2 422 | . . 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 1342 ∈ wcel 2135 ⊆ wss 3111 (class class class)co 5836 ℝcr 7743 + caddc 7747 [,]cicc 9818 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-icc 9822 |
This theorem is referenced by: (None) |
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