Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem15 Structured version   Visualization version   GIF version

Theorem stoweidlem15 45029
Description: This lemma is used to prove the existence of a function 𝑝 as in Lemma 1 from [BrosowskiDeutsh] p. 90: 𝑝 is in the subalgebra, such that 0 ≀ p ≀ 1, p_(t0) = 0, and p > 0 on T - U. Here (πΊβ€˜πΌ) is used to represent p_(ti) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem15.1 𝑄 = {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))}
stoweidlem15.3 (πœ‘ β†’ 𝐺:(1...𝑀)βŸΆπ‘„)
stoweidlem15.4 ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ 𝑓:π‘‡βŸΆβ„)
Assertion
Ref Expression
stoweidlem15 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ (((πΊβ€˜πΌ)β€˜π‘†) ∈ ℝ ∧ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘†) ∧ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑓,𝐼   𝑇,𝑓   πœ‘,𝑓   𝑑,β„Ž,𝐺   𝐴,β„Ž   β„Ž,𝐼,𝑑   𝑇,β„Ž,𝑑   β„Ž,𝑍
Allowed substitution hints:   πœ‘(𝑑,β„Ž)   𝐴(𝑑)   𝑄(𝑑,𝑓,β„Ž)   𝑆(𝑑,𝑓,β„Ž)   𝑀(𝑑,𝑓,β„Ž)   𝑍(𝑑,𝑓)

Proof of Theorem stoweidlem15
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 simpl 481 . . . 4 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ πœ‘)
2 stoweidlem15.3 . . . . . 6 (πœ‘ β†’ 𝐺:(1...𝑀)βŸΆπ‘„)
32ffvelcdmda 7085 . . . . 5 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ (πΊβ€˜πΌ) ∈ 𝑄)
4 elrabi 3676 . . . . . 6 ((πΊβ€˜πΌ) ∈ {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))} β†’ (πΊβ€˜πΌ) ∈ 𝐴)
5 stoweidlem15.1 . . . . . 6 𝑄 = {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))}
64, 5eleq2s 2849 . . . . 5 ((πΊβ€˜πΌ) ∈ 𝑄 β†’ (πΊβ€˜πΌ) ∈ 𝐴)
73, 6syl 17 . . . 4 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ (πΊβ€˜πΌ) ∈ 𝐴)
8 eleq1 2819 . . . . . . . 8 (𝑓 = (πΊβ€˜πΌ) β†’ (𝑓 ∈ 𝐴 ↔ (πΊβ€˜πΌ) ∈ 𝐴))
98anbi2d 627 . . . . . . 7 (𝑓 = (πΊβ€˜πΌ) β†’ ((πœ‘ ∧ 𝑓 ∈ 𝐴) ↔ (πœ‘ ∧ (πΊβ€˜πΌ) ∈ 𝐴)))
10 feq1 6697 . . . . . . 7 (𝑓 = (πΊβ€˜πΌ) β†’ (𝑓:π‘‡βŸΆβ„ ↔ (πΊβ€˜πΌ):π‘‡βŸΆβ„))
119, 10imbi12d 343 . . . . . 6 (𝑓 = (πΊβ€˜πΌ) β†’ (((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ 𝑓:π‘‡βŸΆβ„) ↔ ((πœ‘ ∧ (πΊβ€˜πΌ) ∈ 𝐴) β†’ (πΊβ€˜πΌ):π‘‡βŸΆβ„)))
12 stoweidlem15.4 . . . . . 6 ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ 𝑓:π‘‡βŸΆβ„)
1311, 12vtoclg 3541 . . . . 5 ((πΊβ€˜πΌ) ∈ 𝐴 β†’ ((πœ‘ ∧ (πΊβ€˜πΌ) ∈ 𝐴) β†’ (πΊβ€˜πΌ):π‘‡βŸΆβ„))
147, 13syl 17 . . . 4 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ ((πœ‘ ∧ (πΊβ€˜πΌ) ∈ 𝐴) β†’ (πΊβ€˜πΌ):π‘‡βŸΆβ„))
151, 7, 14mp2and 695 . . 3 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ (πΊβ€˜πΌ):π‘‡βŸΆβ„)
1615ffvelcdmda 7085 . 2 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ ((πΊβ€˜πΌ)β€˜π‘†) ∈ ℝ)
173, 5eleqtrdi 2841 . . . . . . 7 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ (πΊβ€˜πΌ) ∈ {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))})
18 fveq1 6889 . . . . . . . . . 10 (β„Ž = (πΊβ€˜πΌ) β†’ (β„Žβ€˜π‘) = ((πΊβ€˜πΌ)β€˜π‘))
1918eqeq1d 2732 . . . . . . . . 9 (β„Ž = (πΊβ€˜πΌ) β†’ ((β„Žβ€˜π‘) = 0 ↔ ((πΊβ€˜πΌ)β€˜π‘) = 0))
20 fveq1 6889 . . . . . . . . . . . 12 (β„Ž = (πΊβ€˜πΌ) β†’ (β„Žβ€˜π‘‘) = ((πΊβ€˜πΌ)β€˜π‘‘))
2120breq2d 5159 . . . . . . . . . . 11 (β„Ž = (πΊβ€˜πΌ) β†’ (0 ≀ (β„Žβ€˜π‘‘) ↔ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘)))
2220breq1d 5157 . . . . . . . . . . 11 (β„Ž = (πΊβ€˜πΌ) β†’ ((β„Žβ€˜π‘‘) ≀ 1 ↔ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))
2321, 22anbi12d 629 . . . . . . . . . 10 (β„Ž = (πΊβ€˜πΌ) β†’ ((0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1) ↔ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1)))
2423ralbidv 3175 . . . . . . . . 9 (β„Ž = (πΊβ€˜πΌ) β†’ (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1) ↔ βˆ€π‘‘ ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1)))
2519, 24anbi12d 629 . . . . . . . 8 (β„Ž = (πΊβ€˜πΌ) β†’ (((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1)) ↔ (((πΊβ€˜πΌ)β€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))))
2625elrab 3682 . . . . . . 7 ((πΊβ€˜πΌ) ∈ {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))} ↔ ((πΊβ€˜πΌ) ∈ 𝐴 ∧ (((πΊβ€˜πΌ)β€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))))
2717, 26sylib 217 . . . . . 6 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ ((πΊβ€˜πΌ) ∈ 𝐴 ∧ (((πΊβ€˜πΌ)β€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))))
2827simprd 494 . . . . 5 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ (((πΊβ€˜πΌ)β€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1)))
2928simprd 494 . . . 4 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ βˆ€π‘‘ ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))
30 fveq2 6890 . . . . . . . 8 (𝑠 = 𝑑 β†’ ((πΊβ€˜πΌ)β€˜π‘ ) = ((πΊβ€˜πΌ)β€˜π‘‘))
3130breq2d 5159 . . . . . . 7 (𝑠 = 𝑑 β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ↔ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘)))
3230breq1d 5157 . . . . . . 7 (𝑠 = 𝑑 β†’ (((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1 ↔ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))
3331, 32anbi12d 629 . . . . . 6 (𝑠 = 𝑑 β†’ ((0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ∧ ((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1) ↔ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1)))
3433cbvralvw 3232 . . . . 5 (βˆ€π‘  ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ∧ ((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1) ↔ βˆ€π‘‘ ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))
35 fveq2 6890 . . . . . . . 8 (𝑠 = 𝑆 β†’ ((πΊβ€˜πΌ)β€˜π‘ ) = ((πΊβ€˜πΌ)β€˜π‘†))
3635breq2d 5159 . . . . . . 7 (𝑠 = 𝑆 β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ↔ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘†)))
3735breq1d 5157 . . . . . . 7 (𝑠 = 𝑆 β†’ (((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1 ↔ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
3836, 37anbi12d 629 . . . . . 6 (𝑠 = 𝑆 β†’ ((0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ∧ ((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1) ↔ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘†) ∧ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1)))
3938rspccva 3610 . . . . 5 ((βˆ€π‘  ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ∧ ((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1) ∧ 𝑆 ∈ 𝑇) β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘†) ∧ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
4034, 39sylanbr 580 . . . 4 ((βˆ€π‘‘ ∈ 𝑇 (0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘) ∧ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1) ∧ 𝑆 ∈ 𝑇) β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘†) ∧ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
4129, 40sylan 578 . . 3 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘†) ∧ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
4241simpld 493 . 2 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘†))
4341simprd 494 . 2 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1)
4416, 42, 433jca 1126 1 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ (((πΊβ€˜πΌ)β€˜π‘†) ∈ ℝ ∧ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘†) ∧ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430   class class class wbr 5147  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  β„cr 11111  0cc0 11112  1c1 11113   ≀ cle 11253  ...cfz 13488
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550
This theorem is referenced by:  stoweidlem30  45044  stoweidlem38  45052  stoweidlem44  45058
  Copyright terms: Public domain W3C validator