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Theorem stoweidlem16 40551
Description: Lemma for stoweid 40598. 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 1081 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝜑)
3 fveq1 6228 . . . . . . . . . . 11 ( = 𝑓 → (𝑡) = (𝑓𝑡))
43breq2d 4697 . . . . . . . . . 10 ( = 𝑓 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑓𝑡)))
53breq1d 4695 . . . . . . . . . 10 ( = 𝑓 → ((𝑡) ≤ 1 ↔ (𝑓𝑡) ≤ 1))
64, 5anbi12d 747 . . . . . . . . 9 ( = 𝑓 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
76ralbidv 3015 . . . . . . . 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 3399 . . . . . . 7 (𝑓𝑌 ↔ (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
109simplbi 475 . . . . . 6 (𝑓𝑌𝑓𝐴)
11103ad2ant2 1103 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝑓𝐴)
12 fveq1 6228 . . . . . . . . . . 11 ( = 𝑔 → (𝑡) = (𝑔𝑡))
1312breq2d 4697 . . . . . . . . . 10 ( = 𝑔 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑔𝑡)))
1412breq1d 4695 . . . . . . . . . 10 ( = 𝑔 → ((𝑡) ≤ 1 ↔ (𝑔𝑡) ≤ 1))
1513, 14anbi12d 747 . . . . . . . . 9 ( = 𝑔 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1615ralbidv 3015 . . . . . . . 8 ( = 𝑔 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1716, 8elrab2 3399 . . . . . . 7 (𝑔𝑌 ↔ (𝑔𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1817simplbi 475 . . . . . 6 (𝑔𝑌𝑔𝐴)
19183ad2ant3 1104 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝑔𝐴)
20 stoweidlem16.5 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
212, 11, 19, 20syl3anc 1366 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
221, 21syl5eqel 2734 . . 3 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝐴)
23 stoweidlem16.1 . . . . 5 𝑡𝜑
24 nfra1 2970 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
25 nfcv 2793 . . . . . . . 8 𝑡𝐴
2624, 25nfrab 3153 . . . . . . 7 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
278, 26nfcxfr 2791 . . . . . 6 𝑡𝑌
2827nfcri 2787 . . . . 5 𝑡 𝑓𝑌
2927nfcri 2787 . . . . 5 𝑡 𝑔𝑌
3023, 28, 29nf3an 1871 . . . 4 𝑡(𝜑𝑓𝑌𝑔𝑌)
312, 11jca 553 . . . . . . . . . . 11 ((𝜑𝑓𝑌𝑔𝑌) → (𝜑𝑓𝐴))
3231adantr 480 . . . . . . . . . 10 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝜑𝑓𝐴))
33 stoweidlem16.4 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
3432, 33syl 17 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 𝑓:𝑇⟶ℝ)
35 simpr 476 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 𝑡𝑇)
3634, 35ffvelrnd 6400 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ∈ ℝ)
372, 19jca 553 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → (𝜑𝑔𝐴))
38 eleq1 2718 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓𝐴𝑔𝐴))
3938anbi2d 740 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝜑𝑓𝐴) ↔ (𝜑𝑔𝐴)))
40 feq1 6064 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓:𝑇⟶ℝ ↔ 𝑔:𝑇⟶ℝ))
4139, 40imbi12d 333 . . . . . . . . . . 11 (𝑓 = 𝑔 → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ)))
4241, 33vtoclg 3297 . . . . . . . . . 10 (𝑔𝐴 → ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ))
4319, 37, 42sylc 65 . . . . . . . . 9 ((𝜑𝑓𝑌𝑔𝑌) → 𝑔:𝑇⟶ℝ)
4443ffvelrnda 6399 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ∈ ℝ)
459simprbi 479 . . . . . . . . . . 11 (𝑓𝑌 → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
46453ad2ant2 1103 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4746r19.21bi 2961 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4847simpld 474 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑓𝑡))
4917simprbi 479 . . . . . . . . . . 11 (𝑔𝑌 → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
50493ad2ant3 1104 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5150r19.21bi 2961 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5251simpld 474 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑔𝑡))
5336, 44, 48, 52mulge0d 10642 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ ((𝑓𝑡) · (𝑔𝑡)))
5436, 44remulcld 10108 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ)
551fvmpt2 6330 . . . . . . . 8 ((𝑡𝑇 ∧ ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5635, 54, 55syl2anc 694 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5753, 56breqtrrd 4713 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝐻𝑡))
58 1red 10093 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 1 ∈ ℝ)
5947simprd 478 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ≤ 1)
6051simprd 478 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ≤ 1)
6136, 58, 44, 58, 48, 52, 59, 60lemul12ad 11004 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ (1 · 1))
62 1t1e1 11213 . . . . . . . 8 (1 · 1) = 1
6361, 62syl6breq 4726 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ 1)
6456, 63eqbrtrd 4707 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) ≤ 1)
6557, 64jca 553 . . . . 5 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
6665ex 449 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
6730, 66ralrimi 2986 . . 3 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
68 nfmpt1 4780 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
691, 68nfcxfr 2791 . . . . . 6 𝑡𝐻
7069nfeq2 2809 . . . . 5 𝑡 = 𝐻
71 fveq1 6228 . . . . . . 7 ( = 𝐻 → (𝑡) = (𝐻𝑡))
7271breq2d 4697 . . . . . 6 ( = 𝐻 → (0 ≤ (𝑡) ↔ 0 ≤ (𝐻𝑡)))
7371breq1d 4695 . . . . . 6 ( = 𝐻 → ((𝑡) ≤ 1 ↔ (𝐻𝑡) ≤ 1))
7472, 73anbi12d 747 . . . . 5 ( = 𝐻 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7570, 74ralbid 3012 . . . 4 ( = 𝐻 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7675elrab 3396 . . 3 (𝐻 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝐻𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7722, 67, 76sylanbrc 699 . 2 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
7877, 8syl6eleqr 2741 1 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wral 2941  {crab 2945   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   · cmul 9979  cle 10113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307
This theorem is referenced by:  stoweidlem48  40583  stoweidlem51  40586
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