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Theorem stoweidlem16 41170
Description: Lemma for stoweid 41217. 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 1127 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝜑)
3 fveq1 6447 . . . . . . . . . . 11 ( = 𝑓 → (𝑡) = (𝑓𝑡))
43breq2d 4900 . . . . . . . . . 10 ( = 𝑓 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑓𝑡)))
53breq1d 4898 . . . . . . . . . 10 ( = 𝑓 → ((𝑡) ≤ 1 ↔ (𝑓𝑡) ≤ 1))
64, 5anbi12d 624 . . . . . . . . 9 ( = 𝑓 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
76ralbidv 3168 . . . . . . . 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 3576 . . . . . . 7 (𝑓𝑌 ↔ (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
109simplbi 493 . . . . . 6 (𝑓𝑌𝑓𝐴)
11103ad2ant2 1125 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝑓𝐴)
12 fveq1 6447 . . . . . . . . . . 11 ( = 𝑔 → (𝑡) = (𝑔𝑡))
1312breq2d 4900 . . . . . . . . . 10 ( = 𝑔 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑔𝑡)))
1412breq1d 4898 . . . . . . . . . 10 ( = 𝑔 → ((𝑡) ≤ 1 ↔ (𝑔𝑡) ≤ 1))
1513, 14anbi12d 624 . . . . . . . . 9 ( = 𝑔 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1615ralbidv 3168 . . . . . . . 8 ( = 𝑔 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1716, 8elrab2 3576 . . . . . . 7 (𝑔𝑌 ↔ (𝑔𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1817simplbi 493 . . . . . 6 (𝑔𝑌𝑔𝐴)
19183ad2ant3 1126 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝑔𝐴)
20 stoweidlem16.5 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
212, 11, 19, 20syl3anc 1439 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
221, 21syl5eqel 2863 . . 3 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝐴)
23 stoweidlem16.1 . . . . 5 𝑡𝜑
24 nfra1 3123 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
25 nfcv 2934 . . . . . . . 8 𝑡𝐴
2624, 25nfrab 3310 . . . . . . 7 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
278, 26nfcxfr 2932 . . . . . 6 𝑡𝑌
2827nfcri 2929 . . . . 5 𝑡 𝑓𝑌
2927nfcri 2929 . . . . 5 𝑡 𝑔𝑌
3023, 28, 29nf3an 1948 . . . 4 𝑡(𝜑𝑓𝑌𝑔𝑌)
312, 11jca 507 . . . . . . . . . . 11 ((𝜑𝑓𝑌𝑔𝑌) → (𝜑𝑓𝐴))
3231adantr 474 . . . . . . . . . 10 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝜑𝑓𝐴))
33 stoweidlem16.4 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
3432, 33syl 17 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 𝑓:𝑇⟶ℝ)
35 simpr 479 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 𝑡𝑇)
3634, 35ffvelrnd 6626 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ∈ ℝ)
372, 19jca 507 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → (𝜑𝑔𝐴))
38 eleq1w 2842 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓𝐴𝑔𝐴))
3938anbi2d 622 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝜑𝑓𝐴) ↔ (𝜑𝑔𝐴)))
40 feq1 6274 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓:𝑇⟶ℝ ↔ 𝑔:𝑇⟶ℝ))
4139, 40imbi12d 336 . . . . . . . . . . 11 (𝑓 = 𝑔 → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ)))
4241, 33vtoclg 3467 . . . . . . . . . 10 (𝑔𝐴 → ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ))
4319, 37, 42sylc 65 . . . . . . . . 9 ((𝜑𝑓𝑌𝑔𝑌) → 𝑔:𝑇⟶ℝ)
4443ffvelrnda 6625 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ∈ ℝ)
459simprbi 492 . . . . . . . . . . 11 (𝑓𝑌 → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
46453ad2ant2 1125 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4746r19.21bi 3114 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4847simpld 490 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑓𝑡))
4917simprbi 492 . . . . . . . . . . 11 (𝑔𝑌 → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
50493ad2ant3 1126 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5150r19.21bi 3114 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5251simpld 490 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑔𝑡))
5336, 44, 48, 52mulge0d 10954 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ ((𝑓𝑡) · (𝑔𝑡)))
5436, 44remulcld 10409 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ)
551fvmpt2 6554 . . . . . . . 8 ((𝑡𝑇 ∧ ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5635, 54, 55syl2anc 579 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5753, 56breqtrrd 4916 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝐻𝑡))
58 1red 10379 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 1 ∈ ℝ)
5947simprd 491 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ≤ 1)
6051simprd 491 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ≤ 1)
6136, 58, 44, 58, 48, 52, 59, 60lemul12ad 11322 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ (1 · 1))
62 1t1e1 11548 . . . . . . . 8 (1 · 1) = 1
6361, 62syl6breq 4929 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ 1)
6456, 63eqbrtrd 4910 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) ≤ 1)
6557, 64jca 507 . . . . 5 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
6665ex 403 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
6730, 66ralrimi 3139 . . 3 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
68 nfmpt1 4984 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
691, 68nfcxfr 2932 . . . . . 6 𝑡𝐻
7069nfeq2 2949 . . . . 5 𝑡 = 𝐻
71 fveq1 6447 . . . . . . 7 ( = 𝐻 → (𝑡) = (𝐻𝑡))
7271breq2d 4900 . . . . . 6 ( = 𝐻 → (0 ≤ (𝑡) ↔ 0 ≤ (𝐻𝑡)))
7371breq1d 4898 . . . . . 6 ( = 𝐻 → ((𝑡) ≤ 1 ↔ (𝐻𝑡) ≤ 1))
7472, 73anbi12d 624 . . . . 5 ( = 𝐻 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7570, 74ralbid 3165 . . . 4 ( = 𝐻 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7675elrab 3572 . . 3 (𝐻 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝐻𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7722, 67, 76sylanbrc 578 . 2 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
7877, 8syl6eleqr 2870 1 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wnf 1827  wcel 2107  wral 3090  {crab 3094   class class class wbr 4888  cmpt 4967  wf 6133  cfv 6137  (class class class)co 6924  cr 10273  0cc0 10274  1c1 10275   · cmul 10279  cle 10414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-po 5276  df-so 5277  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611
This theorem is referenced by:  stoweidlem48  41202  stoweidlem51  41205
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