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Mirrors > Home > ILE Home > Th. List > lediv2a | 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 | pm3.2 139 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ → (𝐶 ∈ ℝ → (𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ))) | |
2 | 1 | pm2.43i 49 | . . . . . 6 ⊢ (𝐶 ∈ ℝ → (𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ)) |
3 | 2 | adantr 276 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) → (𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ)) |
4 | leid 8039 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ → 𝐶 ≤ 𝐶) | |
5 | 4 | anim2i 342 | . . . . . 6 ⊢ ((0 ≤ 𝐶 ∧ 𝐶 ∈ ℝ) → (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶)) |
6 | 5 | ancoms 268 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) → (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶)) |
7 | 3, 6 | jca 306 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) → ((𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶))) |
8 | 7 | ad2antlr 489 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → ((𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶))) |
9 | 8 | 3adantl2 1154 | . 2 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → ((𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶))) |
10 | id 19 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) | |
11 | 10 | ad2ant2r 509 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
12 | 11 | adantr 276 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ 𝐴 ≤ 𝐵) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
13 | simplr 528 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < 𝐴) | |
14 | 13 | anim1i 340 | . . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ 𝐴 ≤ 𝐵) → (0 < 𝐴 ∧ 𝐴 ≤ 𝐵)) |
15 | 12, 14 | jca 306 | . . 3 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ 𝐴 ≤ 𝐵) → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 𝐴 ≤ 𝐵))) |
16 | 15 | 3adantl3 1155 | . 2 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 𝐴 ≤ 𝐵))) |
17 | lediv12a 8849 | . 2 ⊢ ((((𝐶 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 ≤ 𝐶)) ∧ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 𝐴 ≤ 𝐵))) → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)) | |
18 | 9, 16, 17 | syl2anc 411 | 1 ⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 ∈ wcel 2148 class class class wbr 4003 (class class class)co 5874 ℝcr 7809 0cc0 7810 < clt 7990 ≤ cle 7991 / cdiv 8627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 |
This theorem is referenced by: lediv2ad 9717 |
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