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Theorem stoweidlem15 44346
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 484 . . . 4 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ πœ‘)
2 stoweidlem15.3 . . . . . 6 (πœ‘ β†’ 𝐺:(1...𝑀)βŸΆπ‘„)
32ffvelcdmda 7039 . . . . 5 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ (πΊβ€˜πΌ) ∈ 𝑄)
4 elrabi 3643 . . . . . 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 2852 . . . . 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 2822 . . . . . . . 8 (𝑓 = (πΊβ€˜πΌ) β†’ (𝑓 ∈ 𝐴 ↔ (πΊβ€˜πΌ) ∈ 𝐴))
98anbi2d 630 . . . . . . 7 (𝑓 = (πΊβ€˜πΌ) β†’ ((πœ‘ ∧ 𝑓 ∈ 𝐴) ↔ (πœ‘ ∧ (πΊβ€˜πΌ) ∈ 𝐴)))
10 feq1 6653 . . . . . . 7 (𝑓 = (πΊβ€˜πΌ) β†’ (𝑓:π‘‡βŸΆβ„ ↔ (πΊβ€˜πΌ):π‘‡βŸΆβ„))
119, 10imbi12d 345 . . . . . 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 3527 . . . . 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 698 . . 3 ((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) β†’ (πΊβ€˜πΌ):π‘‡βŸΆβ„)
1615ffvelcdmda 7039 . 2 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ ((πΊβ€˜πΌ)β€˜π‘†) ∈ ℝ)
173, 5eleqtrdi 2844 . . . . . . 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 6845 . . . . . . . . . 10 (β„Ž = (πΊβ€˜πΌ) β†’ (β„Žβ€˜π‘) = ((πΊβ€˜πΌ)β€˜π‘))
1918eqeq1d 2735 . . . . . . . . 9 (β„Ž = (πΊβ€˜πΌ) β†’ ((β„Žβ€˜π‘) = 0 ↔ ((πΊβ€˜πΌ)β€˜π‘) = 0))
20 fveq1 6845 . . . . . . . . . . . 12 (β„Ž = (πΊβ€˜πΌ) β†’ (β„Žβ€˜π‘‘) = ((πΊβ€˜πΌ)β€˜π‘‘))
2120breq2d 5121 . . . . . . . . . . 11 (β„Ž = (πΊβ€˜πΌ) β†’ (0 ≀ (β„Žβ€˜π‘‘) ↔ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘)))
2220breq1d 5119 . . . . . . . . . . 11 (β„Ž = (πΊβ€˜πΌ) β†’ ((β„Žβ€˜π‘‘) ≀ 1 ↔ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))
2321, 22anbi12d 632 . . . . . . . . . 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 3171 . . . . . . . . 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 632 . . . . . . . 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 3649 . . . . . . 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 497 . . . . 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 497 . . . 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 6846 . . . . . . . 8 (𝑠 = 𝑑 β†’ ((πΊβ€˜πΌ)β€˜π‘ ) = ((πΊβ€˜πΌ)β€˜π‘‘))
3130breq2d 5121 . . . . . . 7 (𝑠 = 𝑑 β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ↔ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘‘)))
3230breq1d 5119 . . . . . . 7 (𝑠 = 𝑑 β†’ (((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1 ↔ ((πΊβ€˜πΌ)β€˜π‘‘) ≀ 1))
3331, 32anbi12d 632 . . . . . 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 3224 . . . . 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 6846 . . . . . . . 8 (𝑠 = 𝑆 β†’ ((πΊβ€˜πΌ)β€˜π‘ ) = ((πΊβ€˜πΌ)β€˜π‘†))
3635breq2d 5121 . . . . . . 7 (𝑠 = 𝑆 β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘ ) ↔ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘†)))
3735breq1d 5119 . . . . . . 7 (𝑠 = 𝑆 β†’ (((πΊβ€˜πΌ)β€˜π‘ ) ≀ 1 ↔ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
3836, 37anbi12d 632 . . . . . 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 3582 . . . . 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 583 . . . 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 581 . . 3 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ (0 ≀ ((πΊβ€˜πΌ)β€˜π‘†) ∧ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1))
4241simpld 496 . 2 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ 0 ≀ ((πΊβ€˜πΌ)β€˜π‘†))
4341simprd 497 . 2 (((πœ‘ ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) β†’ ((πΊβ€˜πΌ)β€˜π‘†) ≀ 1)
4416, 42, 433jca 1129 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406   class class class wbr 5109  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  β„cr 11058  0cc0 11059  1c1 11060   ≀ cle 11198  ...cfz 13433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508
This theorem is referenced by:  stoweidlem30  44361  stoweidlem38  44369  stoweidlem44  44375
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