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Mirrors > Home > MPE Home > Th. List > lediv2a | Structured version Visualization version GIF version |
Description: Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) |
Ref | Expression |
---|---|
lediv2a | ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.2 469 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ → (𝐶 ∈ ℝ → (𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ))) | |
2 | 1 | pm2.43i 52 | . . . . . 6 ⊢ (𝐶 ∈ ℝ → (𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ)) |
3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) → (𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ)) |
4 | leid 11311 | . . . . . 6 ⊢ (𝐶 ∈ ℝ → 𝐶 ≤ 𝐶) | |
5 | 4 | anim1ci 615 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) → (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶)) |
6 | 3, 5 | jca 511 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) → ((𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶))) |
7 | 6 | ad2antlr 724 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → ((𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶))) |
8 | 7 | 3adantl2 1164 | . 2 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → ((𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶))) |
9 | id 22 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) | |
10 | 9 | ad2ant2r 744 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
11 | 10 | adantr 480 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ 𝐴 ≤ 𝐵) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
12 | simplr 766 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < 𝐴) | |
13 | 12 | anim1i 614 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ 𝐴 ≤ 𝐵) → (0 < 𝐴 ∧ 𝐴 ≤ 𝐵)) |
14 | 11, 13 | jca 511 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ 𝐴 ≤ 𝐵) → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 𝐴 ≤ 𝐵))) |
15 | 14 | 3adantl3 1165 | . 2 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 𝐴 ≤ 𝐵))) |
16 | lediv12a 12108 | . 2 ⊢ ((((𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶)) ∧ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 𝐴 ≤ 𝐵))) → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)) | |
17 | 8, 15, 16 | syl2anc 583 | 1 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7404 ℝcr 11108 0cc0 11109 < clt 11249 ≤ cle 11250 / cdiv 11872 |
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 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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-rmo 3370 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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 |
This theorem is referenced by: lediv2ad 13041 dchrisum0lem1b 27399 pntrmax 27448 |
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