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Theorem stoweidlem16 40894
Description: Lemma for stoweid 40941. 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 1166 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝜑)
3 fveq1 6378 . . . . . . . . . . 11 ( = 𝑓 → (𝑡) = (𝑓𝑡))
43breq2d 4823 . . . . . . . . . 10 ( = 𝑓 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑓𝑡)))
53breq1d 4821 . . . . . . . . . 10 ( = 𝑓 → ((𝑡) ≤ 1 ↔ (𝑓𝑡) ≤ 1))
64, 5anbi12d 624 . . . . . . . . 9 ( = 𝑓 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
76ralbidv 3133 . . . . . . . 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 3525 . . . . . . 7 (𝑓𝑌 ↔ (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
109simplbi 491 . . . . . 6 (𝑓𝑌𝑓𝐴)
11103ad2ant2 1164 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝑓𝐴)
12 fveq1 6378 . . . . . . . . . . 11 ( = 𝑔 → (𝑡) = (𝑔𝑡))
1312breq2d 4823 . . . . . . . . . 10 ( = 𝑔 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑔𝑡)))
1412breq1d 4821 . . . . . . . . . 10 ( = 𝑔 → ((𝑡) ≤ 1 ↔ (𝑔𝑡) ≤ 1))
1513, 14anbi12d 624 . . . . . . . . 9 ( = 𝑔 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1615ralbidv 3133 . . . . . . . 8 ( = 𝑔 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1716, 8elrab2 3525 . . . . . . 7 (𝑔𝑌 ↔ (𝑔𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1)))
1817simplbi 491 . . . . . 6 (𝑔𝑌𝑔𝐴)
19183ad2ant3 1165 . . . . 5 ((𝜑𝑓𝑌𝑔𝑌) → 𝑔𝐴)
20 stoweidlem16.5 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
212, 11, 19, 20syl3anc 1490 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
221, 21syl5eqel 2848 . . 3 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝐴)
23 stoweidlem16.1 . . . . 5 𝑡𝜑
24 nfra1 3088 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
25 nfcv 2907 . . . . . . . 8 𝑡𝐴
2624, 25nfrab 3271 . . . . . . 7 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
278, 26nfcxfr 2905 . . . . . 6 𝑡𝑌
2827nfcri 2901 . . . . 5 𝑡 𝑓𝑌
2927nfcri 2901 . . . . 5 𝑡 𝑔𝑌
3023, 28, 29nf3an 2000 . . . 4 𝑡(𝜑𝑓𝑌𝑔𝑌)
312, 11jca 507 . . . . . . . . . . 11 ((𝜑𝑓𝑌𝑔𝑌) → (𝜑𝑓𝐴))
3231adantr 472 . . . . . . . . . 10 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝜑𝑓𝐴))
33 stoweidlem16.4 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
3432, 33syl 17 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 𝑓:𝑇⟶ℝ)
35 simpr 477 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 𝑡𝑇)
3634, 35ffvelrnd 6554 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ∈ ℝ)
372, 19jca 507 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → (𝜑𝑔𝐴))
38 eleq1w 2827 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓𝐴𝑔𝐴))
3938anbi2d 622 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝜑𝑓𝐴) ↔ (𝜑𝑔𝐴)))
40 feq1 6206 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓:𝑇⟶ℝ ↔ 𝑔:𝑇⟶ℝ))
4139, 40imbi12d 335 . . . . . . . . . . 11 (𝑓 = 𝑔 → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ)))
4241, 33vtoclg 3418 . . . . . . . . . 10 (𝑔𝐴 → ((𝜑𝑔𝐴) → 𝑔:𝑇⟶ℝ))
4319, 37, 42sylc 65 . . . . . . . . 9 ((𝜑𝑓𝑌𝑔𝑌) → 𝑔:𝑇⟶ℝ)
4443ffvelrnda 6553 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ∈ ℝ)
459simprbi 490 . . . . . . . . . . 11 (𝑓𝑌 → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
46453ad2ant2 1164 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4746r19.21bi 3079 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1))
4847simpld 488 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑓𝑡))
4917simprbi 490 . . . . . . . . . . 11 (𝑔𝑌 → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
50493ad2ant3 1165 . . . . . . . . . 10 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5150r19.21bi 3079 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝑔𝑡) ∧ (𝑔𝑡) ≤ 1))
5251simpld 488 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝑔𝑡))
5336, 44, 48, 52mulge0d 10862 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ ((𝑓𝑡) · (𝑔𝑡)))
5436, 44remulcld 10328 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ)
551fvmpt2 6484 . . . . . . . 8 ((𝑡𝑇 ∧ ((𝑓𝑡) · (𝑔𝑡)) ∈ ℝ) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5635, 54, 55syl2anc 579 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) = ((𝑓𝑡) · (𝑔𝑡)))
5753, 56breqtrrd 4839 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 0 ≤ (𝐻𝑡))
58 1red 10298 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → 1 ∈ ℝ)
5947simprd 489 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑓𝑡) ≤ 1)
6051simprd 489 . . . . . . . . 9 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝑔𝑡) ≤ 1)
6136, 58, 44, 58, 48, 52, 59, 60lemul12ad 11224 . . . . . . . 8 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ (1 · 1))
62 1t1e1 11444 . . . . . . . 8 (1 · 1) = 1
6361, 62syl6breq 4852 . . . . . . 7 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) ≤ 1)
6456, 63eqbrtrd 4833 . . . . . 6 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (𝐻𝑡) ≤ 1)
6557, 64jca 507 . . . . 5 (((𝜑𝑓𝑌𝑔𝑌) ∧ 𝑡𝑇) → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
6665ex 401 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 → (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
6730, 66ralrimi 3104 . . 3 ((𝜑𝑓𝑌𝑔𝑌) → ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1))
68 nfmpt1 4908 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
691, 68nfcxfr 2905 . . . . . 6 𝑡𝐻
7069nfeq2 2923 . . . . 5 𝑡 = 𝐻
71 fveq1 6378 . . . . . . 7 ( = 𝐻 → (𝑡) = (𝐻𝑡))
7271breq2d 4823 . . . . . 6 ( = 𝐻 → (0 ≤ (𝑡) ↔ 0 ≤ (𝐻𝑡)))
7371breq1d 4821 . . . . . 6 ( = 𝐻 → ((𝑡) ≤ 1 ↔ (𝐻𝑡) ≤ 1))
7472, 73anbi12d 624 . . . . 5 ( = 𝐻 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7570, 74ralbid 3130 . . . 4 ( = 𝐻 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝐻𝑡) ∧ (𝐻𝑡) ≤ 1)))
7675elrab 3521 . . 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 2855 1 ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wnf 1878  wcel 2155  wral 3055  {crab 3059   class class class wbr 4811  cmpt 4890  wf 6066  cfv 6070  (class class class)co 6846  cr 10192  0cc0 10193  1c1 10194   · cmul 10198  cle 10333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-po 5200  df-so 5201  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527
This theorem is referenced by:  stoweidlem48  40926  stoweidlem51  40929
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