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Theorem stoweidlem16 43527
Description: Lemma for stoweid 43574. The subset 𝑌 of functions in the algebra 𝐴, with values in [ 0 , 1 ], is closed under multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem16.1 𝑡𝜑
stoweidlem16.2 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
stoweidlem16.3 𝐻 = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
stoweidlem16.4 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem16.5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
Assertion
Ref Expression
stoweidlem16 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝑌)
Distinct variable groups:   𝑓,𝑔,,𝑡,𝐴   𝑇,𝑓,,𝑡   𝜑,𝑓   ,𝐻
Allowed substitution hints:   𝜑(𝑡,𝑔,)   𝑇(𝑔)   𝐻(𝑡,𝑓,𝑔)   𝑌(𝑡,𝑓,𝑔,)

Proof of Theorem stoweidlem16
StepHypRef Expression
1 stoweidlem16.3 . . . 4 𝐻 = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
2 simp1 1135 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝜑)
3 fveq1 6775 . . . . . . . . . . 11 ( = 𝑓 → (𝑡) = (𝑓𝑡))
43breq2d 5088 . . . . . . . . . 10 ( = 𝑓 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑓𝑡)))
53breq1d 5086 . . . . . . . . . 10 ( = 𝑓 → ((𝑡) ≤ 1 ↔ (𝑓𝑡) ≤ 1))
64, 5anbi12d 631 . . . . . . . . 9 ( = 𝑓 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
76ralbidv 3119 . . . . . . . 8 ( = 𝑓 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
8 stoweidlem16.2 . . . . . . . 8 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
97, 8elrab2 3628 . . . . . . 7 (𝑓𝑌 ↔ (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
109simplbi 498 . . . . . 6 (𝑓𝑌𝑓𝐴)
11103ad2ant2 1133 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝑓𝐴)
12 fveq1 6775 . . . . . . . . . . 11 ( = 𝑔 → (𝑡) = (𝑔𝑡))
1312breq2d 5088 . . . . . . . . . 10 ( = 𝑔 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑔𝑡)))
1412breq1d 5086 . . . . . . . . . 10 ( = 𝑔 → ((𝑡) ≤ 1 ↔ (𝑔𝑡) ≤ 1))
1513, 14anbi12d 631 . . . . . . . . 9 ( = 𝑔 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1615ralbidv 3119 . . . . . . . 8 ( = 𝑔 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1716, 8elrab2 3628 . . . . . . 7 (𝑔𝑌 ↔ (𝑔𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1817simplbi 498 . . . . . 6 (𝑔𝑌𝑔𝐴)
19183ad2ant3 1134 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝑔𝐴)
20 stoweidlem16.5 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
212, 11, 19, 20syl3anc 1370 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
221, 21eqeltrid 2843 . . 3 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝐴)
23 stoweidlem16.1 . . . . 5 𝑡𝜑
24 nfra1 3144 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
25 nfcv 2907 . . . . . . . 8 𝑡𝐴
2624, 25nfrabw 3317 . . . . . . 7 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
278, 26nfcxfr 2905 . . . . . 6 𝑡𝑌
2827nfcri 2894 . . . . 5 𝑡 𝑓𝑌
2927nfcri 2894 . . . . 5 𝑡 𝑔𝑌
3023, 28, 29nf3an 1904 . . . 4 𝑡(𝜑𝑓𝑌𝑔𝑌)
312, 11jca 512 . . . . . . . . . . 11 ((𝜑𝑓𝑌𝑔𝑌) → (𝜑𝑓𝐴))
3231adantr 481 . . . . . . . . . 10 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝜑𝑓𝐴))
33 stoweidlem16.4 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
3432, 33syl 17 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 𝑓:𝑇⟶ℝ)
35 simpr 485 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 𝑡𝑇)
3634, 35ffvelrnd 6964 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ∈ ℝ)
372, 19jca 512 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → (𝜑𝑔𝐴))
38 eleq1w 2821 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓𝐴𝑔𝐴))
3938anbi2d 629 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝜑𝑓𝐴) ↔ (𝜑𝑔𝐴)))
40 feq1 6583 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓:𝑇⟶ℝ ↔ 𝑔:𝑇⟶ℝ))
4139, 40imbi12d 345 . . . . . . . . . . 11 (𝑓 = 𝑔 → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ)))
4241, 33vtoclg 3504 . . . . . . . . . 10 (𝑔𝐴 → ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ))
4319, 37, 42sylc 65 . . . . . . . . 9 ((𝜑𝑓𝑌𝑔𝑌) → 𝑔:𝑇⟶ℝ)
4443ffvelrnda 6963 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ∈ ℝ)
459simprbi 497 . . . . . . . . . . 11 (𝑓𝑌 → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
46453ad2ant2 1133 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4746r19.21bi 3134 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4847simpld 495 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑓𝑡))
4917simprbi 497 . . . . . . . . . . 11 (𝑔𝑌 → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
50493ad2ant3 1134 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5150r19.21bi 3134 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5251simpld 495 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑔𝑡))
5336, 44, 48, 52mulge0d 11550 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ ((𝑓𝑡) · (𝑔𝑡)))
5436, 44remulcld 11003 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ)
551fvmpt2 6888 . . . . . . . 8 ((𝑡𝑇 ∧ ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5635, 54, 55syl2anc 584 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5753, 56breqtrrd 5104 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝐻𝑡))
58 1red 10974 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 1 ∈ ℝ)
5947simprd 496 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ≤ 1)
6051simprd 496 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ≤ 1)
6136, 58, 44, 58, 48, 52, 59, 60lemul12ad 11915 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ (1 · 1))
62 1t1e1 12133 . . . . . . . 8 (1 · 1) = 1
6361, 62breqtrdi 5117 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ 1)
6456, 63eqbrtrd 5098 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) ≤ 1)
6557, 64jca 512 . . . . 5 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
6665ex 413 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
6730, 66ralrimi 3141 . . 3 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
68 nfmpt1 5184 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
691, 68nfcxfr 2905 . . . . . 6 𝑡𝐻
7069nfeq2 2924 . . . . 5 𝑡 = 𝐻
71 fveq1 6775 . . . . . . 7 ( = 𝐻 → (𝑡) = (𝐻𝑡))
7271breq2d 5088 . . . . . 6 ( = 𝐻 → (0 ≤ (𝑡) ↔ 0 ≤ (𝐻𝑡)))
7371breq1d 5086 . . . . . 6 ( = 𝐻 → ((𝑡) ≤ 1 ↔ (𝐻𝑡) ≤ 1))
7472, 73anbi12d 631 . . . . 5 ( = 𝐻 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7570, 74ralbid 3160 . . . 4 ( = 𝐻 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7675elrab 3625 . . 3 (𝐻 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝐻𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7722, 67, 76sylanbrc 583 . 2 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
7877, 8eleqtrrdi 2850 1 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wnf 1786  wcel 2106  wral 3064  {crab 3068   class class class wbr 5076  cmpt 5159  wf 6431  cfv 6435  (class class class)co 7277  cr 10868  0cc0 10869  1c1 10870   · cmul 10874  cle 11008
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588  ax-resscn 10926  ax-1cn 10927  ax-icn 10928  ax-addcl 10929  ax-addrcl 10930  ax-mulcl 10931  ax-mulrcl 10932  ax-mulcom 10933  ax-addass 10934  ax-mulass 10935  ax-distr 10936  ax-i2m1 10937  ax-1ne0 10938  ax-1rid 10939  ax-rnegex 10940  ax-rrecex 10941  ax-cnre 10942  ax-pre-lttri 10943  ax-pre-lttrn 10944  ax-pre-ltadd 10945  ax-pre-mulgt0 10946
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-po 5505  df-so 5506  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-riota 7234  df-ov 7280  df-oprab 7281  df-mpo 7282  df-er 8496  df-en 8732  df-dom 8733  df-sdom 8734  df-pnf 11009  df-mnf 11010  df-xr 11011  df-ltxr 11012  df-le 11013  df-sub 11205  df-neg 11206
This theorem is referenced by:  stoweidlem48  43559  stoweidlem51  43562
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