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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 8978 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 416 | . . 3 ⊢ (𝐴[,]𝐵) ⊆ ℝ |
5 | 4 | sseli 2995 | . 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 9016 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))) |
10 | 1, 2, 9 | mpanl12 426 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))) |
11 | 6, 10 | mpan2 415 | . . 3 ⊢ (𝑋 ∈ ℝ → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))) |
12 | 11 | biimpd 142 | . 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 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ⊆ wss 2973 (class class class)co 5532 ℝcr 6980 + caddc 6984 [,]cicc 8914 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-icc 8918 |
This theorem is referenced by: (None) |
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