Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem48 Structured version   Visualization version   GIF version

Theorem stoweidlem48 39593
 Description: This lemma is used to prove that 𝑥 built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x < ε on 𝐴. Here 𝑋 is used to represent 𝑥 in the paper, 𝐸 is used to represent ε in the paper, and 𝐷 is used to represent 𝐴 in the paper (because 𝐴 is always used to represent the subalgebra). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem48.1 𝑖𝜑
stoweidlem48.2 𝑡𝜑
stoweidlem48.3 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
stoweidlem48.4 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
stoweidlem48.5 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
stoweidlem48.6 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
stoweidlem48.7 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
stoweidlem48.8 (𝜑𝑀 ∈ ℕ)
stoweidlem48.9 (𝜑𝑊:(1...𝑀)⟶𝑉)
stoweidlem48.10 (𝜑𝑈:(1...𝑀)⟶𝑌)
stoweidlem48.11 (𝜑𝐷 ran 𝑊)
stoweidlem48.12 (𝜑𝐷𝑇)
stoweidlem48.13 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < 𝐸)
stoweidlem48.14 (𝜑𝑇 ∈ V)
stoweidlem48.15 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem48.16 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem48.17 (𝜑𝐸 ∈ ℝ+)
Assertion
Ref Expression
stoweidlem48 (𝜑 → ∀𝑡𝐷 (𝑋𝑡) < 𝐸)
Distinct variable groups:   𝑓,𝑔,,𝑡,𝐴   𝑓,𝑖,𝑇,,𝑡   𝑓,𝐹,𝑔   𝑓,𝑀,𝑔   𝑈,𝑓,𝑔,,𝑡   𝑓,𝑌,𝑔   𝜑,𝑓,𝑔   𝑇,𝑔   𝐷,𝑖   𝑖,𝐸   𝑖,𝑀   𝑈,𝑖   𝑖,𝑊
Allowed substitution hints:   𝜑(𝑡,,𝑖)   𝐴(𝑖)   𝐷(𝑡,𝑓,𝑔,)   𝑃(𝑡,𝑓,𝑔,,𝑖)   𝐸(𝑡,𝑓,𝑔,)   𝐹(𝑡,,𝑖)   𝑀(𝑡,)   𝑉(𝑡,𝑓,𝑔,,𝑖)   𝑊(𝑡,𝑓,𝑔,)   𝑋(𝑡,𝑓,𝑔,,𝑖)   𝑌(𝑡,,𝑖)   𝑍(𝑡,𝑓,𝑔,,𝑖)

Proof of Theorem stoweidlem48
Dummy variables 𝑗 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem48.2 . 2 𝑡𝜑
2 stoweidlem48.12 . . . . . 6 (𝜑𝐷𝑇)
32sselda 3587 . . . . 5 ((𝜑𝑡𝐷) → 𝑡𝑇)
4 stoweidlem48.1 . . . . . 6 𝑖𝜑
5 stoweidlem48.3 . . . . . . 7 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
6 nfra1 2936 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
7 nfcv 2761 . . . . . . . 8 𝑡𝐴
86, 7nfrab 3115 . . . . . . 7 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
95, 8nfcxfr 2759 . . . . . 6 𝑡𝑌
10 stoweidlem48.4 . . . . . 6 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
11 stoweidlem48.5 . . . . . 6 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
12 stoweidlem48.6 . . . . . 6 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
13 stoweidlem48.7 . . . . . 6 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
14 stoweidlem48.14 . . . . . 6 (𝜑𝑇 ∈ V)
15 stoweidlem48.8 . . . . . 6 (𝜑𝑀 ∈ ℕ)
16 stoweidlem48.10 . . . . . 6 (𝜑𝑈:(1...𝑀)⟶𝑌)
175eleq2i 2690 . . . . . . . . 9 (𝑓𝑌𝑓 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
18 fveq1 6152 . . . . . . . . . . . . 13 ( = 𝑓 → (𝑡) = (𝑓𝑡))
1918breq2d 4630 . . . . . . . . . . . 12 ( = 𝑓 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑓𝑡)))
2018breq1d 4628 . . . . . . . . . . . 12 ( = 𝑓 → ((𝑡) ≤ 1 ↔ (𝑓𝑡) ≤ 1))
2119, 20anbi12d 746 . . . . . . . . . . 11 ( = 𝑓 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2221ralbidv 2981 . . . . . . . . . 10 ( = 𝑓 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2322elrab 3350 . . . . . . . . 9 (𝑓 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2417, 23sylbb 209 . . . . . . . 8 (𝑓𝑌 → (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2524simpld 475 . . . . . . 7 (𝑓𝑌𝑓𝐴)
26 stoweidlem48.15 . . . . . . 7 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
2725, 26sylan2 491 . . . . . 6 ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)
28 eqid 2621 . . . . . . 7 (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
29 stoweidlem48.16 . . . . . . 7 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
301, 5, 28, 26, 29stoweidlem16 39561 . . . . . 6 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
314, 9, 10, 11, 12, 13, 14, 15, 16, 27, 30fmuldfeq 39242 . . . . 5 ((𝜑𝑡𝑇) → (𝑋𝑡) = (𝑍𝑡))
323, 31syldan 487 . . . 4 ((𝜑𝑡𝐷) → (𝑋𝑡) = (𝑍𝑡))
33 elnnuz 11675 . . . . . . . . 9 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ‘1))
3415, 33sylib 208 . . . . . . . 8 (𝜑𝑀 ∈ (ℤ‘1))
3534adantr 481 . . . . . . 7 ((𝜑𝑡𝐷) → 𝑀 ∈ (ℤ‘1))
36 nfv 1840 . . . . . . . . . . . 12 𝑖 𝑡𝑇
374, 36nfan 1825 . . . . . . . . . . 11 𝑖(𝜑𝑡𝑇)
3816ffvelrnda 6320 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
39 fveq1 6152 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑈𝑖) → (𝑡) = ((𝑈𝑖)‘𝑡))
4039breq2d 4630 . . . . . . . . . . . . . . . . . . 19 ( = (𝑈𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑈𝑖)‘𝑡)))
4139breq1d 4628 . . . . . . . . . . . . . . . . . . 19 ( = (𝑈𝑖) → ((𝑡) ≤ 1 ↔ ((𝑈𝑖)‘𝑡) ≤ 1))
4240, 41anbi12d 746 . . . . . . . . . . . . . . . . . 18 ( = (𝑈𝑖) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4342ralbidv 2981 . . . . . . . . . . . . . . . . 17 ( = (𝑈𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4443, 5elrab2 3352 . . . . . . . . . . . . . . . 16 ((𝑈𝑖) ∈ 𝑌 ↔ ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4538, 44sylib 208 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4645simpld 475 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝐴)
47 simpl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
4847, 46jca 554 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈𝑖) ∈ 𝐴))
49 eleq1 2686 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑈𝑖) → (𝑓𝐴 ↔ (𝑈𝑖) ∈ 𝐴))
5049anbi2d 739 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝐴)))
51 feq1 5988 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
5250, 51imbi12d 334 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)))
5352, 26vtoclg 3255 . . . . . . . . . . . . . 14 ((𝑈𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ))
5446, 48, 53sylc 65 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
5554adantlr 750 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
56 simplr 791 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
5755, 56ffvelrnd 6321 . . . . . . . . . . 11 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
58 eqid 2621 . . . . . . . . . . 11 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
5937, 57, 58fmptdf 6348 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)):(1...𝑀)⟶ℝ)
60 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → 𝑡𝑇)
61 ovex 6638 . . . . . . . . . . . . 13 (1...𝑀) ∈ V
62 mptexg 6444 . . . . . . . . . . . . 13 ((1...𝑀) ∈ V → (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V)
6361, 62mp1i 13 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V)
6412fvmpt2 6253 . . . . . . . . . . . 12 ((𝑡𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V) → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
6560, 63, 64syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
6665feq1d 5992 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝐹𝑡):(1...𝑀)⟶ℝ ↔ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)):(1...𝑀)⟶ℝ))
6759, 66mpbird 247 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐹𝑡):(1...𝑀)⟶ℝ)
683, 67syldan 487 . . . . . . . 8 ((𝜑𝑡𝐷) → (𝐹𝑡):(1...𝑀)⟶ℝ)
6968ffvelrnda 6320 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
70 remulcl 9972 . . . . . . . 8 ((𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑘 · 𝑗) ∈ ℝ)
7170adantl 482 . . . . . . 7 (((𝜑𝑡𝐷) ∧ (𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑘 · 𝑗) ∈ ℝ)
7235, 69, 71seqcl 12768 . . . . . 6 ((𝜑𝑡𝐷) → (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ)
7313fvmpt2 6253 . . . . . 6 ((𝑡𝑇 ∧ (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
743, 72, 73syl2anc 692 . . . . 5 ((𝜑𝑡𝐷) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
75 nfcv 2761 . . . . . . . . 9 𝑖𝑇
76 nfmpt1 4712 . . . . . . . . 9 𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
7775, 76nfmpt 4711 . . . . . . . 8 𝑖(𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
7812, 77nfcxfr 2759 . . . . . . 7 𝑖𝐹
79 nfcv 2761 . . . . . . 7 𝑖𝑡
8078, 79nffv 6160 . . . . . 6 𝑖(𝐹𝑡)
81 nfv 1840 . . . . . . 7 𝑖 𝑡𝐷
824, 81nfan 1825 . . . . . 6 𝑖(𝜑𝑡𝐷)
83 nfcv 2761 . . . . . 6 𝑗seq1( · , (𝐹𝑡))
84 eqid 2621 . . . . . 6 seq1( · , (𝐹𝑡)) = seq1( · , (𝐹𝑡))
8515adantr 481 . . . . . 6 ((𝜑𝑡𝐷) → 𝑀 ∈ ℕ)
86 simpll 789 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
87 simpr 477 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
883adantr 481 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
8945simprd 479 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1))
9089r19.21bi 2927 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1))
9190simpld 475 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → 0 ≤ ((𝑈𝑖)‘𝑡))
9286, 87, 88, 91syl21anc 1322 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝑈𝑖)‘𝑡))
9365fveq1d 6155 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
9486, 88, 93syl2anc 692 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
9586, 88, 87, 57syl21anc 1322 . . . . . . . . 9 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
9658fvmpt2 6253 . . . . . . . . 9 ((𝑖 ∈ (1...𝑀) ∧ ((𝑈𝑖)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
9787, 95, 96syl2anc 692 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
9894, 97eqtrd 2655 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑈𝑖)‘𝑡))
9992, 98breqtrrd 4646 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝐹𝑡)‘𝑖))
10090simprd 479 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → ((𝑈𝑖)‘𝑡) ≤ 1)
10186, 87, 88, 100syl21anc 1322 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ≤ 1)
10298, 101eqbrtrd 4640 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) ≤ 1)
103 stoweidlem48.17 . . . . . . 7 (𝜑𝐸 ∈ ℝ+)
104103adantr 481 . . . . . 6 ((𝜑𝑡𝐷) → 𝐸 ∈ ℝ+)
105 stoweidlem48.11 . . . . . . . . . . 11 (𝜑𝐷 ran 𝑊)
106105sselda 3587 . . . . . . . . . 10 ((𝜑𝑡𝐷) → 𝑡 ran 𝑊)
107 eluni 4410 . . . . . . . . . 10 (𝑡 ran 𝑊 ↔ ∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊))
108106, 107sylib 208 . . . . . . . . 9 ((𝜑𝑡𝐷) → ∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊))
109 stoweidlem48.9 . . . . . . . . . . . . . . . 16 (𝜑𝑊:(1...𝑀)⟶𝑉)
110 ffn 6007 . . . . . . . . . . . . . . . 16 (𝑊:(1...𝑀)⟶𝑉𝑊 Fn (1...𝑀))
111 fvelrnb 6205 . . . . . . . . . . . . . . . 16 (𝑊 Fn (1...𝑀) → (𝑤 ∈ ran 𝑊 ↔ ∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤))
112109, 110, 1113syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝑤 ∈ ran 𝑊 ↔ ∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤))
113112biimpa 501 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤)
114113adantrl 751 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡𝑤𝑤 ∈ ran 𝑊)) → ∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤)
115 simplr 791 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑤) ∧ (𝑊𝑗) = 𝑤) → 𝑡𝑤)
116 simpr 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑤) ∧ (𝑊𝑗) = 𝑤) → (𝑊𝑗) = 𝑤)
117115, 116eleqtrrd 2701 . . . . . . . . . . . . . . . 16 (((𝜑𝑡𝑤) ∧ (𝑊𝑗) = 𝑤) → 𝑡 ∈ (𝑊𝑗))
118117ex 450 . . . . . . . . . . . . . . 15 ((𝜑𝑡𝑤) → ((𝑊𝑗) = 𝑤𝑡 ∈ (𝑊𝑗)))
119118reximdv 3011 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑤) → (∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤 → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
120119adantrr 752 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡𝑤𝑤 ∈ ran 𝑊)) → (∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤 → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
121114, 120mpd 15 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝑤𝑤 ∈ ran 𝑊)) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗))
122121ex 450 . . . . . . . . . . 11 (𝜑 → ((𝑡𝑤𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
123122exlimdv 1858 . . . . . . . . . 10 (𝜑 → (∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
124123adantr 481 . . . . . . . . 9 ((𝜑𝑡𝐷) → (∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
125108, 124mpd 15 . . . . . . . 8 ((𝜑𝑡𝐷) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗))
126 simplll 797 . . . . . . . . . . 11 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → 𝜑)
127 simplr 791 . . . . . . . . . . 11 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → 𝑗 ∈ (1...𝑀))
128 simpr 477 . . . . . . . . . . 11 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → 𝑡 ∈ (𝑊𝑗))
129 nfv 1840 . . . . . . . . . . . . . 14 𝑖 𝑗 ∈ (1...𝑀)
130 nfv 1840 . . . . . . . . . . . . . 14 𝑖 𝑡 ∈ (𝑊𝑗)
1314, 129, 130nf3an 1828 . . . . . . . . . . . . 13 𝑖(𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗))
132 nfv 1840 . . . . . . . . . . . . 13 𝑖((𝑈𝑗)‘𝑡) < 𝐸
133131, 132nfim 1822 . . . . . . . . . . . 12 𝑖((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)
134 eleq1 2686 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑖 ∈ (1...𝑀) ↔ 𝑗 ∈ (1...𝑀)))
135 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑊𝑖) = (𝑊𝑗))
136135eleq2d 2684 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑡 ∈ (𝑊𝑖) ↔ 𝑡 ∈ (𝑊𝑗)))
137134, 1363anbi23d 1399 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑖)) ↔ (𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗))))
138 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑈𝑖) = (𝑈𝑗))
139138fveq1d 6155 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑈𝑖)‘𝑡) = ((𝑈𝑗)‘𝑡))
140139breq1d 4628 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑈𝑖)‘𝑡) < 𝐸 ↔ ((𝑈𝑗)‘𝑡) < 𝐸))
141137, 140imbi12d 334 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸) ↔ ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)))
142 stoweidlem48.13 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < 𝐸)
143142r19.21bi 2927 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
1441433impa 1256 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
145133, 141, 144chvar 2261 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)
146126, 127, 128, 145syl3anc 1323 . . . . . . . . . 10 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)
147146ex 450 . . . . . . . . 9 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊𝑗) → ((𝑈𝑗)‘𝑡) < 𝐸))
148147reximdva 3012 . . . . . . . 8 ((𝜑𝑡𝐷) → (∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗) → ∃𝑗 ∈ (1...𝑀)((𝑈𝑗)‘𝑡) < 𝐸))
149125, 148mpd 15 . . . . . . 7 ((𝜑𝑡𝐷) → ∃𝑗 ∈ (1...𝑀)((𝑈𝑗)‘𝑡) < 𝐸)
15082, 129nfan 1825 . . . . . . . . . . . 12 𝑖((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀))
151 nfcv 2761 . . . . . . . . . . . . . 14 𝑖𝑗
15280, 151nffv 6160 . . . . . . . . . . . . 13 𝑖((𝐹𝑡)‘𝑗)
153152nfeq1 2774 . . . . . . . . . . . 12 𝑖((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡)
154150, 153nfim 1822 . . . . . . . . . . 11 𝑖(((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡))
155134anbi2d 739 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀))))
156 fveq2 6153 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘𝑗))
157156, 139eqeq12d 2636 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (((𝐹𝑡)‘𝑖) = ((𝑈𝑖)‘𝑡) ↔ ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡)))
158155, 157imbi12d 334 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑈𝑖)‘𝑡)) ↔ (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡))))
159154, 158, 98chvar 2261 . . . . . . . . . 10 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡))
160159breq1d 4628 . . . . . . . . 9 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝐹𝑡)‘𝑗) < 𝐸 ↔ ((𝑈𝑗)‘𝑡) < 𝐸))
161160biimprd 238 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝑈𝑗)‘𝑡) < 𝐸 → ((𝐹𝑡)‘𝑗) < 𝐸))
162161reximdva 3012 . . . . . . 7 ((𝜑𝑡𝐷) → (∃𝑗 ∈ (1...𝑀)((𝑈𝑗)‘𝑡) < 𝐸 → ∃𝑗 ∈ (1...𝑀)((𝐹𝑡)‘𝑗) < 𝐸))
163149, 162mpd 15 . . . . . 6 ((𝜑𝑡𝐷) → ∃𝑗 ∈ (1...𝑀)((𝐹𝑡)‘𝑗) < 𝐸)
16480, 82, 83, 84, 85, 68, 99, 102, 104, 163fmul01lt1 39245 . . . . 5 ((𝜑𝑡𝐷) → (seq1( · , (𝐹𝑡))‘𝑀) < 𝐸)
16574, 164eqbrtrd 4640 . . . 4 ((𝜑𝑡𝐷) → (𝑍𝑡) < 𝐸)
16632, 165eqbrtrd 4640 . . 3 ((𝜑𝑡𝐷) → (𝑋𝑡) < 𝐸)
167166ex 450 . 2 (𝜑 → (𝑡𝐷 → (𝑋𝑡) < 𝐸))
1681, 167ralrimi 2952 1 (𝜑 → ∀𝑡𝐷 (𝑋𝑡) < 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701  Ⅎwnf 1705   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  {crab 2911  Vcvv 3189   ⊆ wss 3559  ∪ cuni 4407   class class class wbr 4618   ↦ cmpt 4678  ran crn 5080   Fn wfn 5847  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610   ↦ cmpt2 6612  ℝcr 9886  0cc0 9887  1c1 9888   · cmul 9892   < clt 10025   ≤ cle 10026  ℕcn 10971  ℤ≥cuz 11638  ℝ+crp 11783  ...cfz 12275  seqcseq 12748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-n0 11244  df-z 11329  df-uz 11639  df-rp 11784  df-fz 12276  df-fzo 12414  df-seq 12749 This theorem is referenced by:  stoweidlem51  39596
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