lelttrdi - Metamath Proof Explorer

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Theorem lelttrdi 11416

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.)

**Hypotheses**
Ref	Expression
lelttrdi.r	⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ))
lelttrdi.l	⊢ (𝜑 → 𝐵 ≤ 𝐶)

**Assertion**
Ref	Expression
lelttrdi	⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶))

**Proof of Theorem lelttrdi**
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 479	. . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ)
4	1	simp2d 1140	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ)
6	1	simp3d 1141	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
7	6	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐶 ∈ ℝ)
8		simpr 483	. . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
9		lelttrdi.l	. . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
10	9	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐶)
11	3, 5, 7, 8, 10	ltletrd 11414	. 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐶)
12	11	ex 411	1 ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5152 ℝcr 11147 < clt 11288 ≤ cle 11289

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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-resscn 11205 ax-pre-lttri 11222 ax-pre-lttrn 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294

This theorem is referenced by: difgtsumgt 12565 subfzo0 13796 eucrctshift 30081

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