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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrpxdivcld | Structured version Visualization version GIF version |
Description: Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
Ref | Expression |
---|---|
xrpxdivcld.1 | ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) |
xrpxdivcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
xrpxdivcld | ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7368 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 /𝑒 𝐵) = (0 /𝑒 𝐵)) | |
2 | xrpxdivcld.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | xdiv0rp 31842 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (0 /𝑒 𝐵) = 0) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (0 /𝑒 𝐵) = 0) |
5 | 1, 4 | sylan9eqr 2795 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → (𝐴 /𝑒 𝐵) = 0) |
6 | elxrge02 31844 | . . . . 5 ⊢ ((𝐴 /𝑒 𝐵) ∈ (0[,]+∞) ↔ ((𝐴 /𝑒 𝐵) = 0 ∨ (𝐴 /𝑒 𝐵) ∈ ℝ+ ∨ (𝐴 /𝑒 𝐵) = +∞)) | |
7 | 6 | biimpri 227 | . . . 4 ⊢ (((𝐴 /𝑒 𝐵) = 0 ∨ (𝐴 /𝑒 𝐵) ∈ ℝ+ ∨ (𝐴 /𝑒 𝐵) = +∞) → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
8 | 7 | 3o1cs 31441 | . . 3 ⊢ ((𝐴 /𝑒 𝐵) = 0 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
9 | 5, 8 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 0) → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
10 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ+) | |
11 | 2 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → 𝐵 ∈ ℝ+) |
12 | 10, 11 | rpxdivcld 31846 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → (𝐴 /𝑒 𝐵) ∈ ℝ+) |
13 | 7 | 3o2cs 31442 | . . 3 ⊢ ((𝐴 /𝑒 𝐵) ∈ ℝ+ → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
14 | 12, 13 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
15 | oveq1 7368 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 /𝑒 𝐵) = (+∞ /𝑒 𝐵)) | |
16 | xdivpnfrp 31845 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (+∞ /𝑒 𝐵) = +∞) | |
17 | 2, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (+∞ /𝑒 𝐵) = +∞) |
18 | 15, 17 | sylan9eqr 2795 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = +∞) → (𝐴 /𝑒 𝐵) = +∞) |
19 | 7 | 3o3cs 31443 | . . 3 ⊢ ((𝐴 /𝑒 𝐵) = +∞ → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
20 | 18, 19 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝐴 = +∞) → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
21 | xrpxdivcld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
22 | elxrge02 31844 | . . 3 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞)) | |
23 | 21, 22 | sylib 217 | . 2 ⊢ (𝜑 → (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞)) |
24 | 9, 14, 20, 23 | mpjao3dan 1432 | 1 ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 (class class class)co 7361 0cc0 11059 +∞cpnf 11194 ℝ+crp 12923 [,]cicc 13276 /𝑒 cxdiv 31829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xmul 13043 df-ioo 13277 df-ico 13279 df-icc 13280 df-xdiv 31830 |
This theorem is referenced by: measdivcst 32887 measdivcstALTV 32888 |
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