Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > blcom | GIF version |
Description: Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) |
Ref | Expression |
---|---|
blcom | ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elbl2 12780 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅)) | |
2 | elbl3 12782 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋)) → (𝑃 ∈ (𝐴(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅)) | |
3 | 2 | ancom2s 556 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝑃 ∈ (𝐴(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅)) |
4 | 1, 3 | bitr4d 190 | 1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2128 class class class wbr 3965 ‘cfv 5169 (class class class)co 5821 ℝ*cxr 7906 < clt 7907 ∞Metcxmet 12367 ballcbl 12369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-1re 7821 ax-addrcl 7824 ax-0id 7835 ax-rnegex 7836 ax-pre-ltirr 7839 ax-pre-apti 7842 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-map 6592 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-xadd 9675 df-psmet 12374 df-xmet 12375 df-bl 12377 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |