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Mirrors > Home > MPE Home > Th. List > lesub3d | Structured version Visualization version GIF version |
Description: The result of subtracting a number less than or equal to an intermediate number from a number greater than or equal to a third number increased by the intermediate number is greater than or equal to the third number. (Contributed by AV, 13-Aug-2020.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lesub3d.x | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
lesub3d.g | ⊢ (𝜑 → (𝑋 + 𝐶) ≤ 𝐴) |
lesub3d.l | ⊢ (𝜑 → 𝐵 ≤ 𝑋) |
Ref | Expression |
---|---|
lesub3d | ⊢ (𝜑 → 𝐶 ≤ (𝐴 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltadd1d.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
2 | ltnegd.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1, 2 | readdcld 10658 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐵) ∈ ℝ) |
4 | lesub3d.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
5 | 4, 1 | readdcld 10658 | . . 3 ⊢ (𝜑 → (𝑋 + 𝐶) ∈ ℝ) |
6 | leidd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | 1 | recnd 10657 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
8 | 2 | recnd 10657 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | 7, 8 | addcomd 10830 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐵) = (𝐵 + 𝐶)) |
10 | lesub3d.l | . . . . 5 ⊢ (𝜑 → 𝐵 ≤ 𝑋) | |
11 | 2, 4, 1, 10 | leadd1dd 11242 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐶) ≤ (𝑋 + 𝐶)) |
12 | 9, 11 | eqbrtrd 5079 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐵) ≤ (𝑋 + 𝐶)) |
13 | lesub3d.g | . . 3 ⊢ (𝜑 → (𝑋 + 𝐶) ≤ 𝐴) | |
14 | 3, 5, 6, 12, 13 | letrd 10785 | . 2 ⊢ (𝜑 → (𝐶 + 𝐵) ≤ 𝐴) |
15 | leaddsub 11104 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐶 + 𝐵) ≤ 𝐴 ↔ 𝐶 ≤ (𝐴 − 𝐵))) | |
16 | 1, 2, 6, 15 | syl3anc 1363 | . 2 ⊢ (𝜑 → ((𝐶 + 𝐵) ≤ 𝐴 ↔ 𝐶 ≤ (𝐴 − 𝐵))) |
17 | 14, 16 | mpbid 233 | 1 ⊢ (𝜑 → 𝐶 ≤ (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 + caddc 10528 ≤ cle 10664 − cmin 10858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 |
This theorem is referenced by: prmgaplem8 16382 bcmono 25780 fltnlta 39153 |
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