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Mirrors > Home > MPE Home > Th. List > ltaddsub | Structured version Visualization version GIF version |
Description: 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.) |
Ref | Expression |
---|---|
ltaddsub | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lesubadd 11690 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐶 − 𝐵) ≤ 𝐴 ↔ 𝐶 ≤ (𝐴 + 𝐵))) | |
2 | 1 | 3com13 1121 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 − 𝐵) ≤ 𝐴 ↔ 𝐶 ≤ (𝐴 + 𝐵))) |
3 | resubcl 11528 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ) | |
4 | lenlt 11296 | . . . . 5 ⊢ (((𝐶 − 𝐵) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐶 − 𝐵) ≤ 𝐴 ↔ ¬ 𝐴 < (𝐶 − 𝐵))) | |
5 | 3, 4 | stoic3 1770 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐶 − 𝐵) ≤ 𝐴 ↔ ¬ 𝐴 < (𝐶 − 𝐵))) |
6 | 5 | 3com13 1121 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 − 𝐵) ≤ 𝐴 ↔ ¬ 𝐴 < (𝐶 − 𝐵))) |
7 | readdcl 11195 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
8 | lenlt 11296 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ) → (𝐶 ≤ (𝐴 + 𝐵) ↔ ¬ (𝐴 + 𝐵) < 𝐶)) | |
9 | 7, 8 | sylan2 592 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → (𝐶 ≤ (𝐴 + 𝐵) ↔ ¬ (𝐴 + 𝐵) < 𝐶)) |
10 | 9 | 3impb 1112 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ≤ (𝐴 + 𝐵) ↔ ¬ (𝐴 + 𝐵) < 𝐶)) |
11 | 10 | 3coml 1124 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ≤ (𝐴 + 𝐵) ↔ ¬ (𝐴 + 𝐵) < 𝐶)) |
12 | 2, 6, 11 | 3bitr3rd 310 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (¬ (𝐴 + 𝐵) < 𝐶 ↔ ¬ 𝐴 < (𝐶 − 𝐵))) |
13 | 12 | con4bid 317 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7405 ℝcr 11111 + caddc 11115 < clt 11252 ≤ cle 11253 − cmin 11448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 |
This theorem is referenced by: ltaddsub2 11693 ltsub13 11699 ltaddsubi 11779 ltaddsubd 11818 iooshf 13409 ltdifltdiv 13805 swrdswrd 14661 sincosq3sgn 26390 sincosq4sgn 26391 pthdlem1 29532 crctcshwlkn0lem4 29576 breprexplemc 34173 ftc1anclem6 37079 sbgoldbwt 47014 evengpop3 47035 |
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