![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ltadd2 | Structured version Visualization version GIF version |
Description: Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltadd2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axltadd 11332 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵))) | |
2 | oveq2 7439 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐶 + 𝐴) = (𝐶 + 𝐵)) | |
3 | 2 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 = 𝐵 → (𝐶 + 𝐴) = (𝐶 + 𝐵))) |
4 | axltadd 11332 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐴 → (𝐶 + 𝐵) < (𝐶 + 𝐴))) | |
5 | 4 | 3com12 1122 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐴 → (𝐶 + 𝐵) < (𝐶 + 𝐴))) |
6 | 3, 5 | orim12d 966 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ∨ (𝐶 + 𝐵) < (𝐶 + 𝐴)))) |
7 | 6 | con3d 152 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (¬ ((𝐶 + 𝐴) = (𝐶 + 𝐵) ∨ (𝐶 + 𝐵) < (𝐶 + 𝐴)) → ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) |
8 | simp3 1137 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
9 | simp1 1135 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ) | |
10 | 8, 9 | readdcld 11288 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 + 𝐴) ∈ ℝ) |
11 | simp2 1136 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ) | |
12 | 8, 11 | readdcld 11288 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 + 𝐵) ∈ ℝ) |
13 | axlttri 11330 | . . . 4 ⊢ (((𝐶 + 𝐴) ∈ ℝ ∧ (𝐶 + 𝐵) ∈ ℝ) → ((𝐶 + 𝐴) < (𝐶 + 𝐵) ↔ ¬ ((𝐶 + 𝐴) = (𝐶 + 𝐵) ∨ (𝐶 + 𝐵) < (𝐶 + 𝐴)))) | |
14 | 10, 12, 13 | syl2anc 584 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) < (𝐶 + 𝐵) ↔ ¬ ((𝐶 + 𝐴) = (𝐶 + 𝐵) ∨ (𝐶 + 𝐵) < (𝐶 + 𝐴)))) |
15 | axlttri 11330 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) | |
16 | 9, 11, 15 | syl2anc 584 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) |
17 | 7, 14, 16 | 3imtr4d 294 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) < (𝐶 + 𝐵) → 𝐴 < 𝐵)) |
18 | 1, 17 | impbid 212 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 + caddc 11156 < clt 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-addrcl 11214 ax-pre-lttri 11227 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 |
This theorem is referenced by: ltadd2i 11390 ltadd2d 11415 readdcan 11433 ltaddneg 11475 ltadd1 11728 ltaddpos 11751 ltaddsublt 11888 avglt1 12502 flbi2 13854 dp2ltc 32854 sn-ltaddpos 42448 sn-ltaddneg 42449 |
Copyright terms: Public domain | W3C validator |