![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > le2sub | GIF version |
Description: Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.) |
Ref | Expression |
---|---|
le2sub | ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 496 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐴 ∈ ℝ) | |
2 | simprl 498 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐶 ∈ ℝ) | |
3 | simplr 497 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐵 ∈ ℝ) | |
4 | lesub1 7704 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐶 ↔ (𝐴 − 𝐵) ≤ (𝐶 − 𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1170 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (𝐴 ≤ 𝐶 ↔ (𝐴 − 𝐵) ≤ (𝐶 − 𝐵))) |
6 | simprr 499 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐷 ∈ ℝ) | |
7 | lesub2 7705 | . . . 4 ⊢ ((𝐷 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐷 ≤ 𝐵 ↔ (𝐶 − 𝐵) ≤ (𝐶 − 𝐷))) | |
8 | 6, 3, 2, 7 | syl3anc 1170 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (𝐷 ≤ 𝐵 ↔ (𝐶 − 𝐵) ≤ (𝐶 − 𝐷))) |
9 | 5, 8 | anbi12d 457 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐵) ∧ (𝐶 − 𝐵) ≤ (𝐶 − 𝐷)))) |
10 | resubcl 7516 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | |
11 | 10 | adantr 270 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (𝐴 − 𝐵) ∈ ℝ) |
12 | 2, 3 | resubcld 7629 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (𝐶 − 𝐵) ∈ ℝ) |
13 | resubcl 7516 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶 − 𝐷) ∈ ℝ) | |
14 | 13 | adantl 271 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (𝐶 − 𝐷) ∈ ℝ) |
15 | letr 7338 | . . 3 ⊢ (((𝐴 − 𝐵) ∈ ℝ ∧ (𝐶 − 𝐵) ∈ ℝ ∧ (𝐶 − 𝐷) ∈ ℝ) → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐵) ∧ (𝐶 − 𝐵) ≤ (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))) | |
16 | 11, 12, 14, 15 | syl3anc 1170 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐵) ∧ (𝐶 − 𝐵) ≤ (𝐶 − 𝐷)) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))) |
17 | 9, 16 | sylbid 148 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1434 class class class wbr 3806 (class class class)co 5565 ℝcr 7119 ≤ cle 7293 − cmin 7423 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-cnex 7206 ax-resscn 7207 ax-1cn 7208 ax-icn 7210 ax-addcl 7211 ax-addrcl 7212 ax-mulcl 7213 ax-addcom 7215 ax-addass 7217 ax-distr 7219 ax-i2m1 7220 ax-0id 7223 ax-rnegex 7224 ax-cnre 7226 ax-pre-ltwlin 7228 ax-pre-ltadd 7231 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2613 df-sbc 2826 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-br 3807 df-opab 3861 df-id 4077 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-iota 4918 df-fun 4955 df-fv 4961 df-riota 5521 df-ov 5568 df-oprab 5569 df-mpt2 5570 df-pnf 7294 df-mnf 7295 df-xr 7296 df-ltxr 7297 df-le 7298 df-sub 7425 df-neg 7426 |
This theorem is referenced by: le2subd 7808 |
Copyright terms: Public domain | W3C validator |