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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 8409 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))) |
6 | 1, 5 | mpbid 147 | 1 ⊢ (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5897 ℝcr 7841 + caddc 7845 < clt 8023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-i2m1 7947 ax-0id 7950 ax-rnegex 7951 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-iota 5196 df-fv 5243 df-ov 5900 df-pnf 8025 df-mnf 8026 df-ltxr 8028 |
This theorem is referenced by: zltaddlt1le 10039 rebtwn2zlemstep 10285 rebtwn2z 10287 2tnp1ge0ge0 10334 cvg1nlemcau 11028 resqrexlemdec 11055 cos12dec 11810 eirraplem 11819 ivthinclemlopn 14591 cosq23lt0 14731 cosordlem 14747 |
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