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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xdivcld | Structured version Visualization version GIF version |
Description: Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.) |
Ref | Expression |
---|---|
xdivcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xdivcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
xdivcld.3 | ⊢ (𝜑 → 𝐵 ≠ 0) |
Ref | Expression |
---|---|
xdivcld | ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xdivcld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xdivcld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | xdivcld.3 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 0) | |
4 | xdivval 32888 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1373 | . 2 ⊢ (𝜑 → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)) |
6 | xreceu 32891 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) | |
7 | 1, 2, 3, 6 | syl3anc 1373 | . . 3 ⊢ (𝜑 → ∃!𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) |
8 | riotacl 7403 | . . 3 ⊢ (∃!𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴 → (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ ℝ*) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ ℝ*) |
10 | 5, 9 | eqeltrd 2840 | 1 ⊢ (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ∃!wreu 3377 ℩crio 7385 (class class class)co 7429 ℝcr 11150 0cc0 11151 ℝ*cxr 11290 ·e cxmu 13149 /𝑒 cxdiv 32886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-po 5590 df-so 5591 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-1st 8010 df-2nd 8011 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-xneg 13150 df-xmul 13152 df-xdiv 32887 |
This theorem is referenced by: xdivcl 32893 xdivrec 32896 |
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