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Mirrors > Home > MPE Home > Th. List > 1div0 | Structured version Visualization version GIF version |
Description: You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that ∅ is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1div0 | ⊢ (1 / 0) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-div 11033 | . . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
2 | riotaex 6887 | . . 3 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V | |
3 | 1, 2 | dmmpt2 7520 | . 2 ⊢ dom / = (ℂ × (ℂ ∖ {0})) |
4 | eqid 2778 | . . 3 ⊢ 0 = 0 | |
5 | eldifsni 4553 | . . . . 5 ⊢ (0 ∈ (ℂ ∖ {0}) → 0 ≠ 0) | |
6 | 5 | adantl 475 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) → 0 ≠ 0) |
7 | 6 | necon2bi 2999 | . . 3 ⊢ (0 = 0 → ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) |
8 | 4, 7 | ax-mp 5 | . 2 ⊢ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) |
9 | ndmovg 7094 | . 2 ⊢ ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅) | |
10 | 3, 8, 9 | mp2an 682 | 1 ⊢ (1 / 0) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 ∅c0 4141 {csn 4398 × cxp 5353 dom cdm 5355 ℩crio 6882 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 · cmul 10277 / cdiv 11032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-div 11033 |
This theorem is referenced by: (None) |
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