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Mirrors > Home > MPE Home > Th. List > lelttrdi | Structured version Visualization version GIF version |
Description: If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.) |
Ref | Expression |
---|---|
lelttrdi.r | ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) |
lelttrdi.l | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
lelttrdi | ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lelttrdi.r | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) | |
2 | 1 | simp1d 1139 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ) |
4 | 1 | simp2d 1140 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ) |
6 | 1 | simp3d 1141 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐶 ∈ ℝ) |
8 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
9 | lelttrdi.l | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐶) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐶) |
11 | 3, 5, 7, 8, 10 | ltletrd 11371 | . 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐶) |
12 | 11 | ex 412 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5138 ℝcr 11105 < clt 11245 ≤ cle 11246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 |
This theorem is referenced by: difgtsumgt 12522 subfzo0 13751 eucrctshift 29965 |
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