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Theorem stoweidlem15 44721
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 483 . . . 4 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ πœ‘)
2 stoweidlem15.3 . . . . . 6 (πœ‘ β†’ 𝐺:(1...𝑀)βŸΆπ‘„)
32ffvelcdmda 7086 . . . . 5 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ (πΊβ€˜πΌ) ∈ 𝑄)
4 elrabi 3677 . . . . . 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 2851 . . . . 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 2821 . . . . . . . 8 (𝑓 = (πΊβ€˜πΌ) β†’ (𝑓 ∈ 𝐴 ↔ (πΊβ€˜πΌ) ∈ 𝐴))
98anbi2d 629 . . . . . . 7 (𝑓 = (πΊβ€˜πΌ) β†’ ((πœ‘ ∧ 𝑓 ∈ 𝐴) ↔ (πœ‘ ∧ (πΊβ€˜πΌ) ∈ 𝐴)))
10 feq1 6698 . . . . . . 7 (𝑓 = (πΊβ€˜πΌ) β†’ (𝑓:π‘‡βŸΆβ„ ↔ (πΊβ€˜πΌ):π‘‡βŸΆβ„))
119, 10imbi12d 344 . . . . . 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 3556 . . . . 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 697 . . 3 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ (πΊβ€˜πΌ):π‘‡βŸΆβ„)
1615ffvelcdmda 7086 . 2 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ ((πΊβ€˜πΌ)β€˜π‘†) ∈ ℝ)
173, 5eleqtrdi 2843 . . . . . . 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 6890 . . . . . . . . . 10 (β„Ž = (πΊβ€˜πΌ) β†’ (β„Žβ€˜π‘) = ((πΊβ€˜πΌ)β€˜π‘))
1918eqeq1d 2734 . . . . . . . . 9 (β„Ž = (πΊβ€˜πΌ) β†’ ((β„Žβ€˜π‘) = 0 ↔ ((πΊβ€˜πΌ)β€˜π‘) = 0))
20 fveq1 6890 . . . . . . . . . . . 12 (β„Ž = (πΊβ€˜πΌ) β†’ (β„Žβ€˜π‘‘) = ((πΊβ€˜πΌ)β€˜π‘‘))
2120breq2d 5160 . . . . . . . . . . 11 (β„Ž = (πΊβ€˜πΌ) β†’ (0 ≀ (β„Žβ€˜π‘‘) ↔ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘)))
2220breq1d 5158 . . . . . . . . . . 11 (β„Ž = (πΊβ€˜πΌ) β†’ ((β„Žβ€˜π‘‘) ≀ 1 ↔ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))
2321, 22anbi12d 631 . . . . . . . . . 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 3177 . . . . . . . . 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 631 . . . . . . . 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 3683 . . . . . . 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 496 . . . . 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 496 . . . 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 6891 . . . . . . . 8 (𝑠 = 𝑑 β†’ ((πΊβ€˜πΌ)β€˜π‘ ) = ((πΊβ€˜πΌ)β€˜π‘‘))
3130breq2d 5160 . . . . . . 7 (𝑠 = 𝑑 β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ↔ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘)))
3230breq1d 5158 . . . . . . 7 (𝑠 = 𝑑 β†’ (((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1 ↔ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))
3331, 32anbi12d 631 . . . . . 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 3234 . . . . 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 6891 . . . . . . . 8 (𝑠 = 𝑆 β†’ ((πΊβ€˜πΌ)β€˜π‘ ) = ((πΊβ€˜πΌ)β€˜π‘†))
3635breq2d 5160 . . . . . . 7 (𝑠 = 𝑆 β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ↔ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘†)))
3735breq1d 5158 . . . . . . 7 (𝑠 = 𝑆 β†’ (((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1 ↔ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
3836, 37anbi12d 631 . . . . . 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 3611 . . . . 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 582 . . . 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 580 . . 3 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘†) ∧ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
4241simpld 495 . 2 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘†))
4341simprd 496 . 2 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1)
4416, 42, 433jca 1128 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   class class class wbr 5148  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  β„cr 11108  0cc0 11109  1c1 11110   ≀ cle 11248  ...cfz 13483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by:  stoweidlem30  44736  stoweidlem38  44744  stoweidlem44  44750
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