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Theorem stoweidlem16 45972
Description: Lemma for stoweid 46019. 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 6906 . . . . . . . . . . 11 ( = 𝑓 → (𝑡) = (𝑓𝑡))
43breq2d 5160 . . . . . . . . . 10 ( = 𝑓 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑓𝑡)))
53breq1d 5158 . . . . . . . . . 10 ( = 𝑓 → ((𝑡) ≤ 1 ↔ (𝑓𝑡) ≤ 1))
64, 5anbi12d 632 . . . . . . . . 9 ( = 𝑓 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
76ralbidv 3176 . . . . . . . 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 3698 . . . . . . 7 (𝑓𝑌 ↔ (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
109simplbi 497 . . . . . 6 (𝑓𝑌𝑓𝐴)
11103ad2ant2 1133 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝑓𝐴)
12 fveq1 6906 . . . . . . . . . . 11 ( = 𝑔 → (𝑡) = (𝑔𝑡))
1312breq2d 5160 . . . . . . . . . 10 ( = 𝑔 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑔𝑡)))
1412breq1d 5158 . . . . . . . . . 10 ( = 𝑔 → ((𝑡) ≤ 1 ↔ (𝑔𝑡) ≤ 1))
1513, 14anbi12d 632 . . . . . . . . 9 ( = 𝑔 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1615ralbidv 3176 . . . . . . . 8 ( = 𝑔 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1716, 8elrab2 3698 . . . . . . 7 (𝑔𝑌 ↔ (𝑔𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1817simplbi 497 . . . . . 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 3282 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
25 nfcv 2903 . . . . . . . 8 𝑡𝐴
2624, 25nfrabw 3473 . . . . . . 7 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
278, 26nfcxfr 2901 . . . . . 6 𝑡𝑌
2827nfcri 2895 . . . . 5 𝑡 𝑓𝑌
2927nfcri 2895 . . . . 5 𝑡 𝑔𝑌
3023, 28, 29nf3an 1899 . . . 4 𝑡(𝜑𝑓𝑌𝑔𝑌)
312, 11jca 511 . . . . . . . . . . 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 484 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 𝑡𝑇)
3634, 35ffvelcdmd 7105 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ∈ ℝ)
372, 19jca 511 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → (𝜑𝑔𝐴))
38 eleq1w 2822 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓𝐴𝑔𝐴))
3938anbi2d 630 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝜑𝑓𝐴) ↔ (𝜑𝑔𝐴)))
40 feq1 6717 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓:𝑇⟶ℝ ↔ 𝑔:𝑇⟶ℝ))
4139, 40imbi12d 344 . . . . . . . . . . 11 (𝑓 = 𝑔 → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ)))
4241, 33vtoclg 3554 . . . . . . . . . 10 (𝑔𝐴 → ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ))
4319, 37, 42sylc 65 . . . . . . . . 9 ((𝜑𝑓𝑌𝑔𝑌) → 𝑔:𝑇⟶ℝ)
4443ffvelcdmda 7104 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ∈ ℝ)
459simprbi 496 . . . . . . . . . . 11 (𝑓𝑌 → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
46453ad2ant2 1133 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4746r19.21bi 3249 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4847simpld 494 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑓𝑡))
4917simprbi 496 . . . . . . . . . . 11 (𝑔𝑌 → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
50493ad2ant3 1134 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5150r19.21bi 3249 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5251simpld 494 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑔𝑡))
5336, 44, 48, 52mulge0d 11838 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ ((𝑓𝑡) · (𝑔𝑡)))
5436, 44remulcld 11289 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ)
551fvmpt2 7027 . . . . . . . 8 ((𝑡𝑇 ∧ ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5635, 54, 55syl2anc 584 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5753, 56breqtrrd 5176 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝐻𝑡))
58 1red 11260 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 1 ∈ ℝ)
5947simprd 495 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ≤ 1)
6051simprd 495 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ≤ 1)
6136, 58, 44, 58, 48, 52, 59, 60lemul12ad 12208 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ (1 · 1))
62 1t1e1 12426 . . . . . . . 8 (1 · 1) = 1
6361, 62breqtrdi 5189 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ 1)
6456, 63eqbrtrd 5170 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) ≤ 1)
6557, 64jca 511 . . . . 5 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
6665ex 412 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
6730, 66ralrimi 3255 . . 3 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
68 nfmpt1 5256 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
691, 68nfcxfr 2901 . . . . . 6 𝑡𝐻
7069nfeq2 2921 . . . . 5 𝑡 = 𝐻
71 fveq1 6906 . . . . . . 7 ( = 𝐻 → (𝑡) = (𝐻𝑡))
7271breq2d 5160 . . . . . 6 ( = 𝐻 → (0 ≤ (𝑡) ↔ 0 ≤ (𝐻𝑡)))
7371breq1d 5158 . . . . . 6 ( = 𝐻 → ((𝑡) ≤ 1 ↔ (𝐻𝑡) ≤ 1))
7472, 73anbi12d 632 . . . . 5 ( = 𝐻 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7570, 74ralbid 3271 . . . 4 ( = 𝐻 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7675elrab 3695 . . 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 395  w3a 1086   = wceq 1537  wnf 1780  wcel 2106  wral 3059  {crab 3433   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153  1c1 11154   · cmul 11158  cle 11294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493
This theorem is referenced by:  stoweidlem48  46004  stoweidlem51  46007
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