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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 1143 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 2 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ) |
4 | 1 | simp2d 1144 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
5 | 4 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ) |
6 | 1 | simp3d 1145 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
7 | 6 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐶 ∈ ℝ) |
8 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
9 | lelttrdi.l | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐶) | |
10 | 9 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐶) |
11 | 3, 5, 7, 8, 10 | ltletrd 11322 | . 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐶) |
12 | 11 | ex 414 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5110 ℝcr 11057 < clt 11196 ≤ cle 11197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 |
This theorem is referenced by: difgtsumgt 12473 subfzo0 13701 eucrctshift 29229 |
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