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Mirrors > Home > ILE Home > Th. List > ltadd2dd | GIF version |
Description: Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
ltadd2d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltadd2d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd2d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ltletrd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
ltadd2dd | ⊢ (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltletrd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | ltadd2d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltadd2d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | ltadd2d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 2, 3, 4 | ltadd2d 7996 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))) |
6 | 1, 5 | mpbid 146 | 1 ⊢ (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1445 class class class wbr 3867 (class class class)co 5690 ℝcr 7446 + caddc 7450 < clt 7619 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-i2m1 7547 ax-0id 7550 ax-rnegex 7551 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-xp 4473 df-iota 5014 df-fv 5057 df-ov 5693 df-pnf 7621 df-mnf 7622 df-ltxr 7624 |
This theorem is referenced by: zltaddlt1le 9572 rebtwn2zlemstep 9813 rebtwn2z 9815 2tnp1ge0ge0 9857 cvg1nlemcau 10532 resqrexlemdec 10559 eirraplem 11213 |
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