Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  neicvgfv Structured version   Visualization version   GIF version

Theorem neicvgfv 38239
Description: The value of the neighborhoods (convergents) in terms of the the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.)
Hypotheses
Ref Expression
neicvg.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
neicvg.p 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
neicvg.d 𝐷 = (𝑃𝐵)
neicvg.f 𝐹 = (𝒫 𝐵𝑂𝐵)
neicvg.g 𝐺 = (𝐵𝑂𝒫 𝐵)
neicvg.h 𝐻 = (𝐹 ∘ (𝐷𝐺))
neicvg.r (𝜑𝑁𝐻𝑀)
neicvgfv.x (𝜑𝑋𝐵)
Assertion
Ref Expression
neicvgfv (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)})
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝐵,𝑛,𝑜,𝑝,𝑠   𝐷,𝑖,𝑗,𝑘,𝑙,𝑚   𝐷,𝑛,𝑜,𝑝   𝑖,𝐹,𝑗,𝑘,𝑙   𝑛,𝐹,𝑜,𝑝   𝑖,𝐺,𝑗,𝑘,𝑙,𝑚   𝑛,𝐺,𝑜,𝑝   𝑖,𝑀,𝑗,𝑘,𝑙   𝑛,𝑀,𝑜,𝑝   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚,𝑠   𝑛,𝑁,𝑜,𝑝   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠   𝜑,𝑛,𝑜,𝑝
Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑠)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝐹(𝑚,𝑠)   𝐺(𝑠)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝑀(𝑚,𝑠)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝑋(𝑖,𝑗,𝑘,𝑛,𝑜,𝑝)

Proof of Theorem neicvgfv
StepHypRef Expression
1 dfin5 3575 . 2 (𝒫 𝐵 ∩ (𝑁𝑋)) = {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)}
2 neicvg.o . . . . . . 7 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 neicvg.p . . . . . . 7 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
4 neicvg.d . . . . . . 7 𝐷 = (𝑃𝐵)
5 neicvg.f . . . . . . 7 𝐹 = (𝒫 𝐵𝑂𝐵)
6 neicvg.g . . . . . . 7 𝐺 = (𝐵𝑂𝒫 𝐵)
7 neicvg.h . . . . . . 7 𝐻 = (𝐹 ∘ (𝐷𝐺))
8 neicvg.r . . . . . . 7 (𝜑𝑁𝐻𝑀)
92, 3, 4, 5, 6, 7, 8neicvgnex 38236 . . . . . 6 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
10 elmapi 7864 . . . . . 6 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
119, 10syl 17 . . . . 5 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
12 neicvgfv.x . . . . 5 (𝜑𝑋𝐵)
1311, 12ffvelrnd 6346 . . . 4 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1413elpwid 4161 . . 3 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
15 sseqin2 3809 . . 3 ((𝑁𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
1614, 15sylib 208 . 2 (𝜑 → (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
178adantr 481 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑁𝐻𝑀)
1812adantr 481 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
19 simpr 477 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
202, 3, 4, 5, 6, 7, 17, 18, 19neicvgel1 38237 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑠) ∈ (𝑀𝑋)))
2120rabbidva 3183 . 2 (𝜑 → {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)})
221, 16, 213eqtr3a 2678 1 (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195  cdif 3564  cin 3566  wss 3567  𝒫 cpw 4149   class class class wbr 4644  cmpt 4720  ccom 5108  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  𝑚 cmap 7842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator