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Mirrors > Home > ILE Home > Th. List > leltaddd | GIF version |
Description: Adding both sides of two orderings. (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) |
leltaddd.5 | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
leltaddd.6 | ⊢ (𝜑 → 𝐵 < 𝐷) |
Ref | Expression |
---|---|
leltaddd | ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leltaddd.5 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
2 | leltaddd.6 | . 2 ⊢ (𝜑 → 𝐵 < 𝐷) | |
3 | leidd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | ltnegd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | ltadd1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | lt2addd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | leltadd 7846 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))) | |
8 | 3, 4, 5, 6, 7 | syl22anc 1173 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))) |
9 | 1, 2, 8 | mp2and 424 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1436 class class class wbr 3814 (class class class)co 5594 ℝcr 7270 + caddc 7274 < clt 7443 ≤ cle 7444 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-addcom 7366 ax-addass 7368 ax-i2m1 7371 ax-0id 7374 ax-rnegex 7375 ax-pre-ltwlin 7379 ax-pre-ltadd 7382 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-rab 2364 df-v 2616 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-br 3815 df-opab 3869 df-xp 4410 df-cnv 4412 df-iota 4937 df-fv 4980 df-ov 5597 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 |
This theorem is referenced by: flqdiv 9631 |
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