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Mirrors > Home > ILE Home > Th. List > lt2sub | GIF version |
Description: Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.) |
Ref | Expression |
---|---|
lt2sub | ⊢ (((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 | ltsub1 7629 | . . . 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 | ltsub2 7630 | . . . 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 7439 | . . . 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 7552 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (𝐶 − 𝐵) ∈ ℝ) |
13 | resubcl 7439 | . . . 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 | lttr 7252 | . . 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 3793 (class class class)co 5543 ℝcr 7042 < clt 7215 − cmin 7346 |
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 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-addcom 7138 ax-addass 7140 ax-distr 7142 ax-i2m1 7143 ax-0id 7146 ax-rnegex 7147 ax-cnre 7149 ax-pre-lttrn 7152 ax-pre-ltadd 7154 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-id 4056 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-iota 4897 df-fun 4934 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-pnf 7217 df-mnf 7218 df-ltxr 7220 df-sub 7348 df-neg 7349 |
This theorem is referenced by: lt2subd 7735 |
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