![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ltsubadd2b | Structured version Visualization version GIF version |
Description: Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.) |
Ref | Expression |
---|---|
ltsubadd2b | ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐷 − 𝐶) < (𝐵 − 𝐴) ↔ (𝐴 + 𝐷) < (𝐵 + 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
2 | 1 | recnd 11224 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℂ) |
3 | 2 | adantr 481 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐵 ∈ ℂ) |
4 | simpl 483 | . . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → 𝐶 ∈ ℝ) | |
5 | 4 | recnd 11224 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → 𝐶 ∈ ℂ) |
6 | 5 | adantl 482 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐶 ∈ ℂ) |
7 | simpl 483 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
8 | 7 | recnd 11224 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
9 | 8 | adantr 481 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐴 ∈ ℂ) |
10 | 3, 6, 9 | addsubd 11574 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐵 + 𝐶) − 𝐴) = ((𝐵 − 𝐴) + 𝐶)) |
11 | 10 | eqcomd 2737 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐵 − 𝐴) + 𝐶) = ((𝐵 + 𝐶) − 𝐴)) |
12 | 11 | breq2d 5153 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (𝐷 < ((𝐵 − 𝐴) + 𝐶) ↔ 𝐷 < ((𝐵 + 𝐶) − 𝐴))) |
13 | simprr 771 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐷 ∈ ℝ) | |
14 | simprl 769 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐶 ∈ ℝ) | |
15 | resubcl 11506 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) | |
16 | 15 | ancoms 459 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) |
17 | 16 | adantr 481 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (𝐵 − 𝐴) ∈ ℝ) |
18 | 13, 14, 17 | ltsubaddd 11792 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐷 − 𝐶) < (𝐵 − 𝐴) ↔ 𝐷 < ((𝐵 − 𝐴) + 𝐶))) |
19 | 7 | adantr 481 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → 𝐴 ∈ ℝ) |
20 | readdcl 11175 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 + 𝐶) ∈ ℝ) | |
21 | 20 | ad2ant2lr 746 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (𝐵 + 𝐶) ∈ ℝ) |
22 | 19, 13, 21 | ltaddsub2d 11797 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + 𝐷) < (𝐵 + 𝐶) ↔ 𝐷 < ((𝐵 + 𝐶) − 𝐴))) |
23 | 12, 18, 22 | 3bitr4d 310 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐷 − 𝐶) < (𝐵 − 𝐴) ↔ (𝐴 + 𝐷) < (𝐵 + 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5141 (class class class)co 7393 ℂcc 11090 ℝcr 11091 + caddc 11095 < clt 11230 − cmin 11426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 |
This theorem is referenced by: dignn0flhalflem1 46949 |
Copyright terms: Public domain | W3C validator |