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Mirrors > Home > ILE Home > Th. List > lt2addd | GIF version |
Description: Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lt2addd.4 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
lt2addd.5 | ⊢ (𝜑 → 𝐴 < 𝐶) |
lt2addd.6 | ⊢ (𝜑 → 𝐵 < 𝐷) |
Ref | Expression |
---|---|
lt2addd | ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltadd1d.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | lt2addd.4 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
5 | lt2addd.5 | . 2 ⊢ (𝜑 → 𝐴 < 𝐶) | |
6 | lt2addd.6 | . . 3 ⊢ (𝜑 → 𝐵 < 𝐷) | |
7 | 2, 4, 6 | ltled 8072 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐷) |
8 | 1, 2, 3, 4, 5, 7 | ltleaddd 8518 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 class class class wbr 4002 (class class class)co 5872 ℝcr 7807 + caddc 7811 < clt 7988 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-xp 4631 df-cnv 4633 df-iota 5177 df-fv 5223 df-ov 5875 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 |
This theorem is referenced by: modaddmodup 10382 modsumfzodifsn 10391 resqrexlemnm 11020 mertenslemi1 11536 |
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