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Theorem stoweidlem48 46004
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 3995 . . . . 5 ((𝜑𝑡𝐷) → 𝑡𝑇)
4 stoweidlem48.1 . . . . . 6 𝑖𝜑
5 stoweidlem48.3 . . . . . . 7 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
6 nfra1 3282 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
7 nfcv 2903 . . . . . . . 8 𝑡𝐴
86, 7nfrabw 3473 . . . . . . 7 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
95, 8nfcxfr 2901 . . . . . 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 2831 . . . . . . . . 9 (𝑓𝑌𝑓 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
18 fveq1 6906 . . . . . . . . . . . . 13 ( = 𝑓 → (𝑡) = (𝑓𝑡))
1918breq2d 5160 . . . . . . . . . . . 12 ( = 𝑓 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑓𝑡)))
2018breq1d 5158 . . . . . . . . . . . 12 ( = 𝑓 → ((𝑡) ≤ 1 ↔ (𝑓𝑡) ≤ 1))
2119, 20anbi12d 632 . . . . . . . . . . 11 ( = 𝑓 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2221ralbidv 3176 . . . . . . . . . 10 ( = 𝑓 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2322elrab 3695 . . . . . . . . 9 (𝑓 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2417, 23sylbb 219 . . . . . . . 8 (𝑓𝑌 → (𝑓𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑓𝑡) ∧ (𝑓𝑡) ≤ 1)))
2524simpld 494 . . . . . . 7 (𝑓𝑌𝑓𝐴)
26 stoweidlem48.15 . . . . . . 7 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
2725, 26sylan2 593 . . . . . 6 ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)
28 eqid 2735 . . . . . . 7 (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
29 stoweidlem48.16 . . . . . . 7 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
301, 5, 28, 26, 29stoweidlem16 45972 . . . . . 6 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
314, 9, 10, 11, 12, 13, 14, 15, 16, 27, 30fmuldfeq 45539 . . . . 5 ((𝜑𝑡𝑇) → (𝑋𝑡) = (𝑍𝑡))
323, 31syldan 591 . . . 4 ((𝜑𝑡𝐷) → (𝑋𝑡) = (𝑍𝑡))
33 elnnuz 12920 . . . . . . . . 9 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ‘1))
3415, 33sylib 218 . . . . . . . 8 (𝜑𝑀 ∈ (ℤ‘1))
3534adantr 480 . . . . . . 7 ((𝜑𝑡𝐷) → 𝑀 ∈ (ℤ‘1))
36 nfv 1912 . . . . . . . . . . . 12 𝑖 𝑡𝑇
374, 36nfan 1897 . . . . . . . . . . 11 𝑖(𝜑𝑡𝑇)
3816ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
39 fveq1 6906 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑈𝑖) → (𝑡) = ((𝑈𝑖)‘𝑡))
4039breq2d 5160 . . . . . . . . . . . . . . . . . . 19 ( = (𝑈𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑈𝑖)‘𝑡)))
4139breq1d 5158 . . . . . . . . . . . . . . . . . . 19 ( = (𝑈𝑖) → ((𝑡) ≤ 1 ↔ ((𝑈𝑖)‘𝑡) ≤ 1))
4240, 41anbi12d 632 . . . . . . . . . . . . . . . . . 18 ( = (𝑈𝑖) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4342ralbidv 3176 . . . . . . . . . . . . . . . . 17 ( = (𝑈𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4443, 5elrab2 3698 . . . . . . . . . . . . . . . 16 ((𝑈𝑖) ∈ 𝑌 ↔ ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4538, 44sylib 218 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
4645simpld 494 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝐴)
47 simpl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
4847, 46jca 511 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈𝑖) ∈ 𝐴))
49 eleq1 2827 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑈𝑖) → (𝑓𝐴 ↔ (𝑈𝑖) ∈ 𝐴))
5049anbi2d 630 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝐴)))
51 feq1 6717 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
5250, 51imbi12d 344 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)))
5352, 26vtoclg 3554 . . . . . . . . . . . . . 14 ((𝑈𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ))
5446, 48, 53sylc 65 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
5554adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
56 simplr 769 . . . . . . . . . . . 12 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
5755, 56ffvelcdmd 7105 . . . . . . . . . . 11 (((𝜑𝑡𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
58 eqid 2735 . . . . . . . . . . 11 (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
5937, 57, 58fmptdf 7137 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)):(1...𝑀)⟶ℝ)
60 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → 𝑡𝑇)
61 ovex 7464 . . . . . . . . . . . . 13 (1...𝑀) ∈ V
62 mptexg 7241 . . . . . . . . . . . . 13 ((1...𝑀) ∈ V → (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V)
6361, 62mp1i 13 . . . . . . . . . . . 12 ((𝜑𝑡𝑇) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)) ∈ V)
6412fvmpt2 7027 . . . . . . . . . . . 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 6721 . . . . . . . . . 10 ((𝜑𝑡𝑇) → ((𝐹𝑡):(1...𝑀)⟶ℝ ↔ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)):(1...𝑀)⟶ℝ))
6759, 66mpbird 257 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝐹𝑡):(1...𝑀)⟶ℝ)
683, 67syldan 591 . . . . . . . 8 ((𝜑𝑡𝐷) → (𝐹𝑡):(1...𝑀)⟶ℝ)
6968ffvelcdmda 7104 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑘 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑘) ∈ ℝ)
70 remulcl 11238 . . . . . . . 8 ((𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑘 · 𝑗) ∈ ℝ)
7170adantl 481 . . . . . . 7 (((𝜑𝑡𝐷) ∧ (𝑘 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑘 · 𝑗) ∈ ℝ)
7235, 69, 71seqcl 14060 . . . . . 6 ((𝜑𝑡𝐷) → (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ)
7313fvmpt2 7027 . . . . . 6 ((𝑡𝑇 ∧ (seq1( · , (𝐹𝑡))‘𝑀) ∈ ℝ) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
743, 72, 73syl2anc 584 . . . . 5 ((𝜑𝑡𝐷) → (𝑍𝑡) = (seq1( · , (𝐹𝑡))‘𝑀))
75 nfcv 2903 . . . . . . . . 9 𝑖𝑇
76 nfmpt1 5256 . . . . . . . . 9 𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡))
7775, 76nfmpt 5255 . . . . . . . 8 𝑖(𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
7812, 77nfcxfr 2901 . . . . . . 7 𝑖𝐹
79 nfcv 2903 . . . . . . 7 𝑖𝑡
8078, 79nffv 6917 . . . . . 6 𝑖(𝐹𝑡)
81 nfv 1912 . . . . . . 7 𝑖 𝑡𝐷
824, 81nfan 1897 . . . . . 6 𝑖(𝜑𝑡𝐷)
83 nfcv 2903 . . . . . 6 𝑗seq1( · , (𝐹𝑡))
84 eqid 2735 . . . . . 6 seq1( · , (𝐹𝑡)) = seq1( · , (𝐹𝑡))
8515adantr 480 . . . . . 6 ((𝜑𝑡𝐷) → 𝑀 ∈ ℕ)
86 simpll 767 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
87 simpr 484 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑀))
883adantr 480 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡𝑇)
8945simprd 495 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1))
9089r19.21bi 3249 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1))
9190simpld 494 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → 0 ≤ ((𝑈𝑖)‘𝑡))
9286, 87, 88, 91syl21anc 838 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝑈𝑖)‘𝑡))
9365fveq1d 6909 . . . . . . . . 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 838 . . . . . . . . 9 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
9658fvmpt2 7027 . . . . . . . . 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 2775 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑈𝑖)‘𝑡))
9992, 98breqtrrd 5176 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝐹𝑡)‘𝑖))
10090simprd 495 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡𝑇) → ((𝑈𝑖)‘𝑡) ≤ 1)
10186, 87, 88, 100syl21anc 838 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈𝑖)‘𝑡) ≤ 1)
10298, 101eqbrtrd 5170 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) ≤ 1)
103 stoweidlem48.17 . . . . . . 7 (𝜑𝐸 ∈ ℝ+)
104103adantr 480 . . . . . 6 ((𝜑𝑡𝐷) → 𝐸 ∈ ℝ+)
105 stoweidlem48.11 . . . . . . . . . . 11 (𝜑𝐷 ran 𝑊)
106105sselda 3995 . . . . . . . . . 10 ((𝜑𝑡𝐷) → 𝑡 ran 𝑊)
107 eluni 4915 . . . . . . . . . 10 (𝑡 ran 𝑊 ↔ ∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊))
108106, 107sylib 218 . . . . . . . . 9 ((𝜑𝑡𝐷) → ∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊))
109 stoweidlem48.9 . . . . . . . . . . . . . . . 16 (𝜑𝑊:(1...𝑀)⟶𝑉)
110 ffn 6737 . . . . . . . . . . . . . . . 16 (𝑊:(1...𝑀)⟶𝑉𝑊 Fn (1...𝑀))
111 fvelrnb 6969 . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤)
114113adantrl 716 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡𝑤𝑤 ∈ ran 𝑊)) → ∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤)
115 simplr 769 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑤) ∧ (𝑊𝑗) = 𝑤) → 𝑡𝑤)
116 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑𝑡𝑤) ∧ (𝑊𝑗) = 𝑤) → (𝑊𝑗) = 𝑤)
117115, 116eleqtrrd 2842 . . . . . . . . . . . . . . . 16 (((𝜑𝑡𝑤) ∧ (𝑊𝑗) = 𝑤) → 𝑡 ∈ (𝑊𝑗))
118117ex 412 . . . . . . . . . . . . . . 15 ((𝜑𝑡𝑤) → ((𝑊𝑗) = 𝑤𝑡 ∈ (𝑊𝑗)))
119118reximdv 3168 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑤) → (∃𝑗 ∈ (1...𝑀)(𝑊𝑗) = 𝑤 → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
120119adantrr 717 . . . . . . . . . . . . 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 412 . . . . . . . . . . 11 (𝜑 → ((𝑡𝑤𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
123122exlimdv 1931 . . . . . . . . . 10 (𝜑 → (∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
124123adantr 480 . . . . . . . . 9 ((𝜑𝑡𝐷) → (∃𝑤(𝑡𝑤𝑤 ∈ ran 𝑊) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗)))
125108, 124mpd 15 . . . . . . . 8 ((𝜑𝑡𝐷) → ∃𝑗 ∈ (1...𝑀)𝑡 ∈ (𝑊𝑗))
126 simplll 775 . . . . . . . . . . 11 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → 𝜑)
127 simplr 769 . . . . . . . . . . 11 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → 𝑗 ∈ (1...𝑀))
128 simpr 484 . . . . . . . . . . 11 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → 𝑡 ∈ (𝑊𝑗))
129 nfv 1912 . . . . . . . . . . . . . 14 𝑖 𝑗 ∈ (1...𝑀)
130 nfv 1912 . . . . . . . . . . . . . 14 𝑖 𝑡 ∈ (𝑊𝑗)
1314, 129, 130nf3an 1899 . . . . . . . . . . . . 13 𝑖(𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗))
132 nfv 1912 . . . . . . . . . . . . 13 𝑖((𝑈𝑗)‘𝑡) < 𝐸
133131, 132nfim 1894 . . . . . . . . . . . 12 𝑖((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)
134 eleq1 2827 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑖 ∈ (1...𝑀) ↔ 𝑗 ∈ (1...𝑀)))
135 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑊𝑖) = (𝑊𝑗))
136135eleq2d 2825 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑡 ∈ (𝑊𝑖) ↔ 𝑡 ∈ (𝑊𝑗)))
137134, 1363anbi23d 1438 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑖)) ↔ (𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗))))
138 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑈𝑖) = (𝑈𝑗))
139138fveq1d 6909 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑈𝑖)‘𝑡) = ((𝑈𝑗)‘𝑡))
140139breq1d 5158 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑈𝑖)‘𝑡) < 𝐸 ↔ ((𝑈𝑗)‘𝑡) < 𝐸))
141137, 140imbi12d 344 . . . . . . . . . . . 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 3249 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
1441433impa 1109 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
145133, 141, 144chvarfv 2238 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)
146126, 127, 128, 145syl3anc 1370 . . . . . . . . . 10 ((((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑗)) → ((𝑈𝑗)‘𝑡) < 𝐸)
147146ex 412 . . . . . . . . 9 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊𝑗) → ((𝑈𝑗)‘𝑡) < 𝐸))
148147reximdva 3166 . . . . . . . 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 1897 . . . . . . . . . . . 12 𝑖((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀))
151 nfcv 2903 . . . . . . . . . . . . . 14 𝑖𝑗
15280, 151nffv 6917 . . . . . . . . . . . . 13 𝑖((𝐹𝑡)‘𝑗)
153152nfeq1 2919 . . . . . . . . . . . 12 𝑖((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡)
154150, 153nfim 1894 . . . . . . . . . . 11 𝑖(((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡))
155134anbi2d 630 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀))))
156 fveq2 6907 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹𝑡)‘𝑖) = ((𝐹𝑡)‘𝑗))
157156, 139eqeq12d 2751 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (((𝐹𝑡)‘𝑖) = ((𝑈𝑖)‘𝑡) ↔ ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡)))
158155, 157imbi12d 344 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((((𝜑𝑡𝐷) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑖) = ((𝑈𝑖)‘𝑡)) ↔ (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡))))
159154, 158, 98chvarfv 2238 . . . . . . . . . 10 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → ((𝐹𝑡)‘𝑗) = ((𝑈𝑗)‘𝑡))
160159breq1d 5158 . . . . . . . . 9 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝐹𝑡)‘𝑗) < 𝐸 ↔ ((𝑈𝑗)‘𝑡) < 𝐸))
161160biimprd 248 . . . . . . . 8 (((𝜑𝑡𝐷) ∧ 𝑗 ∈ (1...𝑀)) → (((𝑈𝑗)‘𝑡) < 𝐸 → ((𝐹𝑡)‘𝑗) < 𝐸))
162161reximdva 3166 . . . . . . 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 45542 . . . . 5 ((𝜑𝑡𝐷) → (seq1( · , (𝐹𝑡))‘𝑀) < 𝐸)
16574, 164eqbrtrd 5170 . . . 4 ((𝜑𝑡𝐷) → (𝑍𝑡) < 𝐸)
16632, 165eqbrtrd 5170 . . 3 ((𝜑𝑡𝐷) → (𝑋𝑡) < 𝐸)
167166ex 412 . 2 (𝜑 → (𝑡𝐷 → (𝑋𝑡) < 𝐸))
1681, 167ralrimi 3255 1 (𝜑 → ∀𝑡𝐷 (𝑋𝑡) < 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wnf 1780  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963   cuni 4912   class class class wbr 5148  cmpt 5231  ran crn 5690   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  cr 11152  0cc0 11153  1c1 11154   · cmul 11158   < clt 11293  cle 11294  cn 12264  cuz 12876  +crp 13032  ...cfz 13544  seqcseq 14039
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  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-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  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-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  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-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  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  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040
This theorem is referenced by:  stoweidlem51  46007
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