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Theorem stoweidlem48 42199
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 3971 . . . . 5 ((𝜑𝑡𝐷) → 𝑡𝑇)
4 stoweidlem48.1 . . . . . 6 𝑖𝜑
5 stoweidlem48.3 . . . . . . 7 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
6 nfra1 3224 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
7 nfcv 2982 . . . . . . . 8 𝑡𝐴
86, 7nfrab 3392 . . . . . . 7 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
95, 8nfcxfr 2980 . . . . . 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 2909 . . . . . . . . 9 (𝑓𝑌𝑓 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
18 fveq1 6666 . . . . . . . . . . . . 13 ( = 𝑓 → (𝑡) = (𝑓𝑡))
1918breq2d 5075 . . . . . . . . . . . 12 ( = 𝑓 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑓𝑡)))
2018breq1d 5073 . . . . . . . . . . . 12 ( = 𝑓 → ((𝑡) ≤ 1 ↔ (𝑓𝑡) ≤ 1))
2119, 20anbi12d 630 . . . . . . . . . . 11 ( = 𝑓 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2221ralbidv 3202 . . . . . . . . . 10 ( = 𝑓 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2322elrab 3684 . . . . . . . . 9 (𝑓 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2417, 23sylbb 220 . . . . . . . 8 (𝑓𝑌 → (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2524simpld 495 . . . . . . 7 (𝑓𝑌𝑓𝐴)
26 stoweidlem48.15 . . . . . . 7 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
2725, 26sylan2 592 . . . . . 6 ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)
28 eqid 2826 . . . . . . 7 (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
29 stoweidlem48.16 . . . . . . 7 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
301, 5, 28, 26, 29stoweidlem16 42167 . . . . . 6 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
314, 9, 10, 11, 12, 13, 14, 15, 16, 27, 30fmuldfeq 41729 . . . . 5 ((𝜑𝑡𝑇) → (𝑋𝑡) = (𝑍𝑡))
323, 31syldan 591 . . . 4 ((𝜑𝑡𝐷) → (𝑋𝑡) = (𝑍𝑡))
33 elnnuz 12271 . . . . . . . . 9 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ‘1))
3415, 33sylib 219 . . . . . . . 8 (𝜑𝑀 ∈ (ℤ‘1))
3534adantr 481 . . . . . . 7 ((𝜑𝑡𝐷) → 𝑀 ∈ (ℤ‘1))
36 nfv 1908 . . . . . . . . . . . 12 𝑖 𝑡𝑇
374, 36nfan 1893 . . . . . . . . . . 11 𝑖(𝜑𝑡𝑇)
3816ffvelrnda 6847 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
39 fveq1 6666 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑈𝑖) → (𝑡) = ((𝑈𝑖)‘𝑡))
4039breq2d 5075 . . . . . . . . . . . . . . . . . . 19 ( = (𝑈𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑈𝑖)‘𝑡)))
4139breq1d 5073 . . . . . . . . . . . . . . . . . . 19 ( = (𝑈𝑖) → ((𝑡) ≤ 1 ↔ ((𝑈𝑖)‘𝑡) ≤ 1))
4240, 41anbi12d 630 . . . . . . . . . . . . . . . . . 18 ( = (𝑈𝑖) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4342ralbidv 3202 . . . . . . . . . . . . . . . . 17 ( = (𝑈𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4443, 5elrab2 3687 . . . . . . . . . . . . . . . 16 ((𝑈𝑖) ∈ 𝑌 ↔ ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4538, 44sylib 219 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4645simpld 495 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝐴)
47 simpl 483 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
4847, 46jca 512 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈𝑖) ∈ 𝐴))
49 eleq1 2905 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑈𝑖) → (𝑓𝐴 ↔ (𝑈𝑖) ∈ 𝐴))
5049anbi2d 628 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝐴)))
51 feq1 6492 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
5250, 51imbi12d 346 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)))
5352, 26vtoclg 3573 . . . . . . . . . . . . . 14 ((𝑈𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ))
5446, 48, 53sylc 65 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
5554adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
56 simplr 765 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
5755, 56ffvelrnd 6848 . . . . . . . . . . 11 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
58 eqid 2826 . . . . . . . . . . 11 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
5937, 57, 58fmptdf 6877 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)):(1...𝑀)⟶ℝ)
60 simpr 485 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → 𝑡𝑇)
61 ovex 7181 . . . . . . . . . . . . 13 (1...𝑀) ∈ V
62 mptexg 6979 . . . . . . . . . . . . 13 ((1...𝑀) ∈ V → (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V)
6361, 62mp1i 13 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V)
6412fvmpt2 6775 . . . . . . . . . . . 12 ((𝑡𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V) → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
6560, 63, 64syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (𝐹𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
6665feq1d 6496 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝐹𝑡):(1...𝑀)⟶ℝ ↔ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)):(1...𝑀)⟶ℝ))
6759, 66mpbird 258 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐹𝑡):(1...𝑀)⟶ℝ)
683, 67syldan 591 . . . . . . . 8 ((𝜑𝑡𝐷) → (𝐹𝑡):(1...𝑀)⟶ℝ)
6968ffvelrnda 6847 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
70 remulcl 10611 . . . . . . . 8 ((𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑘 · 𝑗) ∈ ℝ)
7170adantl 482 . . . . . . 7 (((𝜑𝑡𝐷) ∧ (𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑘 · 𝑗) ∈ ℝ)
7235, 69, 71seqcl 13380 . . . . . 6 ((𝜑𝑡𝐷) → (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ)
7313fvmpt2 6775 . . . . . 6 ((𝑡𝑇 ∧ (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
743, 72, 73syl2anc 584 . . . . 5 ((𝜑𝑡𝐷) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
75 nfcv 2982 . . . . . . . . 9 𝑖𝑇
76 nfmpt1 5161 . . . . . . . . 9 𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
7775, 76nfmpt 5160 . . . . . . . 8 𝑖(𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
7812, 77nfcxfr 2980 . . . . . . 7 𝑖𝐹
79 nfcv 2982 . . . . . . 7 𝑖𝑡
8078, 79nffv 6677 . . . . . 6 𝑖(𝐹𝑡)
81 nfv 1908 . . . . . . 7 𝑖 𝑡𝐷
824, 81nfan 1893 . . . . . 6 𝑖(𝜑𝑡𝐷)
83 nfcv 2982 . . . . . 6 𝑗seq1( · , (𝐹𝑡))
84 eqid 2826 . . . . . 6 seq1( · , (𝐹𝑡)) = seq1( · , (𝐹𝑡))
8515adantr 481 . . . . . 6 ((𝜑𝑡𝐷) → 𝑀 ∈ ℕ)
86 simpll 763 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
87 simpr 485 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
883adantr 481 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
8945simprd 496 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1))
9089r19.21bi 3213 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1))
9190simpld 495 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → 0 ≤ ((𝑈𝑖)‘𝑡))
9286, 87, 88, 91syl21anc 835 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝑈𝑖)‘𝑡))
9365fveq1d 6669 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
9486, 88, 93syl2anc 584 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖))
9586, 88, 87, 57syl21anc 835 . . . . . . . . 9 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
9658fvmpt2 6775 . . . . . . . . 9 ((𝑖 ∈ (1...𝑀) ∧ ((𝑈𝑖)‘𝑡) ∈ ℝ) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
9787, 95, 96syl2anc 584 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))‘𝑖) = ((𝑈𝑖)‘𝑡))
9894, 97eqtrd 2861 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑈𝑖)‘𝑡))
9992, 98breqtrrd 5091 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝐹𝑡)‘𝑖))
10090simprd 496 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → ((𝑈𝑖)‘𝑡) ≤ 1)
10186, 87, 88, 100syl21anc 835 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ≤ 1)
10298, 101eqbrtrd 5085 . . . . . 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 3971 . . . . . . . . . 10 ((𝜑𝑡𝐷) → 𝑡 ran 𝑊)
107 eluni 4840 . . . . . . . . . 10 (𝑡 ran 𝑊 ↔ ∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊))
108106, 107sylib 219 . . . . . . . . 9 ((𝜑𝑡𝐷) → ∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊))
109 stoweidlem48.9 . . . . . . . . . . . . . . . 16 (𝜑𝑊:(1...𝑀)⟶𝑉)
110 ffn 6511 . . . . . . . . . . . . . . . 16 (𝑊:(1...𝑀)⟶𝑉𝑊 Fn (1...𝑀))
111 fvelrnb 6723 . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤)
114113adantrl 712 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡𝑤𝑤 ∈ ran 𝑊)) → ∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤)
115 simplr 765 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑤) ∧ (𝑊𝑗) = 𝑤) → 𝑡𝑤)
116 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑤) ∧ (𝑊𝑗) = 𝑤) → (𝑊𝑗) = 𝑤)
117115, 116eleqtrrd 2921 . . . . . . . . . . . . . . . 16 (((𝜑𝑡𝑤) ∧ (𝑊𝑗) = 𝑤) → 𝑡 ∈ (𝑊𝑗))
118117ex 413 . . . . . . . . . . . . . . 15 ((𝜑𝑡𝑤) → ((𝑊𝑗) = 𝑤𝑡 ∈ (𝑊𝑗)))
119118reximdv 3278 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑤) → (∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤 → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
120119adantrr 713 . . . . . . . . . . . . 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 413 . . . . . . . . . . 11 (𝜑 → ((𝑡𝑤𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
123122exlimdv 1927 . . . . . . . . . 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 771 . . . . . . . . . . 11 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → 𝜑)
127 simplr 765 . . . . . . . . . . 11 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → 𝑗 ∈ (1...𝑀))
128 simpr 485 . . . . . . . . . . 11 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → 𝑡 ∈ (𝑊𝑗))
129 nfv 1908 . . . . . . . . . . . . . 14 𝑖 𝑗 ∈ (1...𝑀)
130 nfv 1908 . . . . . . . . . . . . . 14 𝑖 𝑡 ∈ (𝑊𝑗)
1314, 129, 130nf3an 1895 . . . . . . . . . . . . 13 𝑖(𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗))
132 nfv 1908 . . . . . . . . . . . . 13 𝑖((𝑈𝑗)‘𝑡) < 𝐸
133131, 132nfim 1890 . . . . . . . . . . . 12 𝑖((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)
134 eleq1 2905 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑖 ∈ (1...𝑀) ↔ 𝑗 ∈ (1...𝑀)))
135 fveq2 6667 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑊𝑖) = (𝑊𝑗))
136135eleq2d 2903 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑡 ∈ (𝑊𝑖) ↔ 𝑡 ∈ (𝑊𝑗)))
137134, 1363anbi23d 1432 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑖)) ↔ (𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗))))
138 fveq2 6667 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑈𝑖) = (𝑈𝑗))
139138fveq1d 6669 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑈𝑖)‘𝑡) = ((𝑈𝑗)‘𝑡))
140139breq1d 5073 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑈𝑖)‘𝑡) < 𝐸 ↔ ((𝑈𝑗)‘𝑡) < 𝐸))
141137, 140imbi12d 346 . . . . . . . . . . . 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 3213 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
1441433impa 1104 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
145133, 141, 144chvar 2409 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)
146126, 127, 128, 145syl3anc 1365 . . . . . . . . . 10 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)
147146ex 413 . . . . . . . . 9 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊𝑗) → ((𝑈𝑗)‘𝑡) < 𝐸))
148147reximdva 3279 . . . . . . . 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 1893 . . . . . . . . . . . 12 𝑖((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀))
151 nfcv 2982 . . . . . . . . . . . . . 14 𝑖𝑗
15280, 151nffv 6677 . . . . . . . . . . . . 13 𝑖((𝐹𝑡)‘𝑗)
153152nfeq1 2998 . . . . . . . . . . . 12 𝑖((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡)
154150, 153nfim 1890 . . . . . . . . . . 11 𝑖(((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡))
155134anbi2d 628 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀))))
156 fveq2 6667 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘𝑗))
157156, 139eqeq12d 2842 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (((𝐹𝑡)‘𝑖) = ((𝑈𝑖)‘𝑡) ↔ ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡)))
158155, 157imbi12d 346 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑈𝑖)‘𝑡)) ↔ (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡))))
159154, 158, 98chvar 2409 . . . . . . . . . 10 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡))
160159breq1d 5073 . . . . . . . . 9 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝐹𝑡)‘𝑗) < 𝐸 ↔ ((𝑈𝑗)‘𝑡) < 𝐸))
161160biimprd 249 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝑈𝑗)‘𝑡) < 𝐸 → ((𝐹𝑡)‘𝑗) < 𝐸))
162161reximdva 3279 . . . . . . 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 41732 . . . . 5 ((𝜑𝑡𝐷) → (seq1( · , (𝐹𝑡))‘𝑀) < 𝐸)
16574, 164eqbrtrd 5085 . . . 4 ((𝜑𝑡𝐷) → (𝑍𝑡) < 𝐸)
16632, 165eqbrtrd 5085 . . 3 ((𝜑𝑡𝐷) → (𝑋𝑡) < 𝐸)
167166ex 413 . 2 (𝜑 → (𝑡𝐷 → (𝑋𝑡) < 𝐸))
1681, 167ralrimi 3221 1 (𝜑 → ∀𝑡𝐷 (𝑋𝑡) < 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wnf 1777  wcel 2107  wral 3143  wrex 3144  {crab 3147  Vcvv 3500  wss 3940   cuni 4837   class class class wbr 5063  cmpt 5143  ran crn 5555   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  cmpo 7150  cr 10525  0cc0 10526  1c1 10527   · cmul 10531   < clt 10664  cle 10665  cn 11627  cuz 12232  +crp 12379  ...cfz 12882  seqcseq 13359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-seq 13360
This theorem is referenced by:  stoweidlem51  42202
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