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 7165 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 /𝑒 𝐵) = (0 /𝑒 𝐵)) | |
2 | xrpxdivcld.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | xdiv0rp 30608 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (0 /𝑒 𝐵) = 0) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (0 /𝑒 𝐵) = 0) |
5 | 1, 4 | sylan9eqr 2880 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 0) → (𝐴 /𝑒 𝐵) = 0) |
6 | elxrge02 30610 | . . . . 5 ⊢ ((𝐴 /𝑒 𝐵) ∈ (0[,]+∞) ↔ ((𝐴 /𝑒 𝐵) = 0 ∨ (𝐴 /𝑒 𝐵) ∈ ℝ+ ∨ (𝐴 /𝑒 𝐵) = +∞)) | |
7 | 6 | biimpri 230 | . . . 4 ⊢ (((𝐴 /𝑒 𝐵) = 0 ∨ (𝐴 /𝑒 𝐵) ∈ ℝ+ ∨ (𝐴 /𝑒 𝐵) = +∞) → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
8 | 7 | 3o1cs 30229 | . . 3 ⊢ ((𝐴 /𝑒 𝐵) = 0 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
9 | 5, 8 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 0) → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
10 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ+) | |
11 | 2 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → 𝐵 ∈ ℝ+) |
12 | 10, 11 | rpxdivcld 30612 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → (𝐴 /𝑒 𝐵) ∈ ℝ+) |
13 | 7 | 3o2cs 30230 | . . 3 ⊢ ((𝐴 /𝑒 𝐵) ∈ ℝ+ → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
14 | 12, 13 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
15 | oveq1 7165 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 /𝑒 𝐵) = (+∞ /𝑒 𝐵)) | |
16 | xdivpnfrp 30611 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (+∞ /𝑒 𝐵) = +∞) | |
17 | 2, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (+∞ /𝑒 𝐵) = +∞) |
18 | 15, 17 | sylan9eqr 2880 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = +∞) → (𝐴 /𝑒 𝐵) = +∞) |
19 | 7 | 3o3cs 30231 | . . 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 30610 | . . 3 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞)) | |
23 | 21, 22 | sylib 220 | . 2 ⊢ (𝜑 → (𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞)) |
24 | 9, 14, 20, 23 | mpjao3dan 1427 | 1 ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 ℝ+crp 12392 [,]cicc 12744 /𝑒 cxdiv 30595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-xdiv 30596 |
This theorem is referenced by: measdivcst 31485 measdivcstALTV 31486 |
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