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| Mirrors > Home > MPE Home > Th. List > Mathboxes > xdivval | Structured version Visualization version GIF version | ||
| Description: Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.) |
| Ref | Expression |
|---|---|
| xdivval | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldifsn 4786 | . . 3 ⊢ (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) | |
| 2 | simpl 482 | . . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → 𝑦 = 𝐴) | |
| 3 | 2 | eqeq2d 2748 | . . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴)) |
| 4 | 3 | riotabidva 7407 | . . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦) = (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴)) |
| 5 | simpl 482 | . . . . . . 7 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → 𝑧 = 𝐵) | |
| 6 | 5 | oveq1d 7446 | . . . . . 6 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥)) |
| 7 | 6 | eqeq1d 2739 | . . . . 5 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴)) |
| 8 | 7 | riotabidva 7407 | . . . 4 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)) |
| 9 | df-xdiv 32900 | . . . 4 ⊢ /𝑒 = (𝑦 ∈ ℝ*, 𝑧 ∈ (ℝ ∖ {0}) ↦ (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦)) | |
| 10 | riotaex 7392 | . . . 4 ⊢ (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ V | |
| 11 | 4, 8, 9, 10 | ovmpo 7593 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)) |
| 12 | 1, 11 | sylan2br 595 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)) |
| 13 | 12 | 3impb 1115 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 ℩crio 7387 (class class class)co 7431 ℝcr 11154 0cc0 11155 ℝ*cxr 11294 ·e cxmu 13153 /𝑒 cxdiv 32899 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xdiv 32900 |
| This theorem is referenced by: xdivcld 32905 xdivmul 32907 rexdiv 32908 xdivpnfrp 32915 |
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