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Theorem stoweidlem27 40717
Description: This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here (𝑞𝑖) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem27.1 𝐺 = (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
stoweidlem27.2 (𝜑𝑄 ∈ V)
stoweidlem27.3 (𝜑𝑀 ∈ ℕ)
stoweidlem27.4 (𝜑𝑌 Fn ran 𝐺)
stoweidlem27.5 (𝜑 → ran 𝐺 ∈ V)
stoweidlem27.6 ((𝜑𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑙)
stoweidlem27.7 (𝜑𝐹:(1...𝑀)–1-1-onto→ran 𝐺)
stoweidlem27.8 (𝜑 → (𝑇𝑈) ⊆ 𝑋)
stoweidlem27.9 𝑡𝜑
stoweidlem27.10 𝑤𝜑
stoweidlem27.11 𝑄
Assertion
Ref Expression
stoweidlem27 (𝜑 → ∃𝑞(𝑀 ∈ ℕ ∧ (𝑞:(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡))))
Distinct variable groups:   ,𝑖,𝑡,𝑤,𝐹   ,𝑙,𝑌,𝑡,𝑤   𝑇,,𝑤   𝑖,𝑞,𝑡,𝐹   𝑖,𝐺   𝑖,𝑀,𝑞   𝑖,𝑋,𝑤   𝑖,𝑌,𝑞   𝜑,𝑖   𝑄,𝑙   𝜑,𝑙   𝐺,𝑙   𝑄,𝑞   𝑇,𝑞   𝑈,𝑞   𝑤,𝑀   𝑤,𝑄   𝑤,𝑈
Allowed substitution hints:   𝜑(𝑤,𝑡,,𝑞)   𝑄(𝑡,,𝑖)   𝑇(𝑡,𝑖,𝑙)   𝑈(𝑡,,𝑖,𝑙)   𝐹(𝑙)   𝐺(𝑤,𝑡,,𝑞)   𝑀(𝑡,,𝑙)   𝑋(𝑡,,𝑞,𝑙)

Proof of Theorem stoweidlem27
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 stoweidlem27.4 . . . 4 (𝜑𝑌 Fn ran 𝐺)
2 stoweidlem27.5 . . . 4 (𝜑 → ran 𝐺 ∈ V)
3 fnex 6700 . . . 4 ((𝑌 Fn ran 𝐺 ∧ ran 𝐺 ∈ V) → 𝑌 ∈ V)
41, 2, 3syl2anc 575 . . 3 (𝜑𝑌 ∈ V)
5 stoweidlem27.7 . . . . 5 (𝜑𝐹:(1...𝑀)–1-1-onto→ran 𝐺)
6 f1ofn 6348 . . . . 5 (𝐹:(1...𝑀)–1-1-onto→ran 𝐺𝐹 Fn (1...𝑀))
75, 6syl 17 . . . 4 (𝜑𝐹 Fn (1...𝑀))
8 ovex 6900 . . . 4 (1...𝑀) ∈ V
9 fnex 6700 . . . 4 ((𝐹 Fn (1...𝑀) ∧ (1...𝑀) ∈ V) → 𝐹 ∈ V)
107, 8, 9sylancl 576 . . 3 (𝜑𝐹 ∈ V)
11 coexg 7341 . . 3 ((𝑌 ∈ V ∧ 𝐹 ∈ V) → (𝑌𝐹) ∈ V)
124, 10, 11syl2anc 575 . 2 (𝜑 → (𝑌𝐹) ∈ V)
13 stoweidlem27.3 . . 3 (𝜑𝑀 ∈ ℕ)
14 f1of 6347 . . . . . 6 (𝐹:(1...𝑀)–1-1-onto→ran 𝐺𝐹:(1...𝑀)⟶ran 𝐺)
155, 14syl 17 . . . . 5 (𝜑𝐹:(1...𝑀)⟶ran 𝐺)
16 fnfco 6278 . . . . 5 ((𝑌 Fn ran 𝐺𝐹:(1...𝑀)⟶ran 𝐺) → (𝑌𝐹) Fn (1...𝑀))
171, 15, 16syl2anc 575 . . . 4 (𝜑 → (𝑌𝐹) Fn (1...𝑀))
18 rncoss 5581 . . . . 5 ran (𝑌𝐹) ⊆ ran 𝑌
19 fvelrnb 6458 . . . . . . . . . . 11 (𝑌 Fn ran 𝐺 → (𝑘 ∈ ran 𝑌 ↔ ∃𝑙 ∈ ran 𝐺(𝑌𝑙) = 𝑘))
201, 19syl 17 . . . . . . . . . 10 (𝜑 → (𝑘 ∈ ran 𝑌 ↔ ∃𝑙 ∈ ran 𝐺(𝑌𝑙) = 𝑘))
2120biimpa 464 . . . . . . . . 9 ((𝜑𝑘 ∈ ran 𝑌) → ∃𝑙 ∈ ran 𝐺(𝑌𝑙) = 𝑘)
22 stoweidlem27.10 . . . . . . . . . . . . . 14 𝑤𝜑
23 stoweidlem27.1 . . . . . . . . . . . . . . . . 17 𝐺 = (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
24 nfmpt1 4934 . . . . . . . . . . . . . . . . 17 𝑤(𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
2523, 24nfcxfr 2942 . . . . . . . . . . . . . . . 16 𝑤𝐺
2625nfrn 5563 . . . . . . . . . . . . . . 15 𝑤ran 𝐺
2726nfcri 2938 . . . . . . . . . . . . . 14 𝑤 𝑙 ∈ ran 𝐺
2822, 27nfan 1990 . . . . . . . . . . . . 13 𝑤(𝜑𝑙 ∈ ran 𝐺)
29 stoweidlem27.6 . . . . . . . . . . . . . . . . 17 ((𝜑𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑙)
3029ad2antrr 708 . . . . . . . . . . . . . . . 16 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → (𝑌𝑙) ∈ 𝑙)
31 simpr 473 . . . . . . . . . . . . . . . 16 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
3230, 31eleqtrd 2883 . . . . . . . . . . . . . . 15 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → (𝑌𝑙) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
33 nfcv 2944 . . . . . . . . . . . . . . . 16 (𝑌𝑙)
34 stoweidlem27.11 . . . . . . . . . . . . . . . 16 𝑄
35 nfv 2005 . . . . . . . . . . . . . . . 16 𝑤 = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)}
36 fveq1 6401 . . . . . . . . . . . . . . . . . . 19 ( = (𝑌𝑙) → (𝑡) = ((𝑌𝑙)‘𝑡))
3736breq2d 4849 . . . . . . . . . . . . . . . . . 18 ( = (𝑌𝑙) → (0 < (𝑡) ↔ 0 < ((𝑌𝑙)‘𝑡)))
3837rabbidv 3375 . . . . . . . . . . . . . . . . 17 ( = (𝑌𝑙) → {𝑡𝑇 ∣ 0 < (𝑡)} = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)})
3938eqeq2d 2812 . . . . . . . . . . . . . . . 16 ( = (𝑌𝑙) → (𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)} ↔ 𝑤 = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)}))
4033, 34, 35, 39elrabf 3551 . . . . . . . . . . . . . . 15 ((𝑌𝑙) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ ((𝑌𝑙) ∈ 𝑄𝑤 = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)}))
4132, 40sylib 209 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ((𝑌𝑙) ∈ 𝑄𝑤 = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)}))
4241simpld 484 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → (𝑌𝑙) ∈ 𝑄)
43 simpr 473 . . . . . . . . . . . . . 14 ((𝜑𝑙 ∈ ran 𝐺) → 𝑙 ∈ ran 𝐺)
4423elrnmpt 5567 . . . . . . . . . . . . . . 15 (𝑙 ∈ ran 𝐺 → (𝑙 ∈ ran 𝐺 ↔ ∃𝑤𝑋 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}))
4543, 44syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑙 ∈ ran 𝐺) → (𝑙 ∈ ran 𝐺 ↔ ∃𝑤𝑋 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}))
4643, 45mpbid 223 . . . . . . . . . . . . 13 ((𝜑𝑙 ∈ ran 𝐺) → ∃𝑤𝑋 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
4728, 42, 46r19.29af 3260 . . . . . . . . . . . 12 ((𝜑𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑄)
4847adantlr 697 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ran 𝑌) ∧ 𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑄)
49 eleq1 2869 . . . . . . . . . . 11 ((𝑌𝑙) = 𝑘 → ((𝑌𝑙) ∈ 𝑄𝑘𝑄))
5048, 49syl5ibcom 236 . . . . . . . . . 10 (((𝜑𝑘 ∈ ran 𝑌) ∧ 𝑙 ∈ ran 𝐺) → ((𝑌𝑙) = 𝑘𝑘𝑄))
5150reximdva 3200 . . . . . . . . 9 ((𝜑𝑘 ∈ ran 𝑌) → (∃𝑙 ∈ ran 𝐺(𝑌𝑙) = 𝑘 → ∃𝑙 ∈ ran 𝐺 𝑘𝑄))
5221, 51mpd 15 . . . . . . . 8 ((𝜑𝑘 ∈ ran 𝑌) → ∃𝑙 ∈ ran 𝐺 𝑘𝑄)
53 idd 24 . . . . . . . . . 10 (𝑙 ∈ ran 𝐺 → (𝑘𝑄𝑘𝑄))
5453a1i 11 . . . . . . . . 9 ((𝜑𝑘 ∈ ran 𝑌) → (𝑙 ∈ ran 𝐺 → (𝑘𝑄𝑘𝑄)))
5554rexlimdv 3214 . . . . . . . 8 ((𝜑𝑘 ∈ ran 𝑌) → (∃𝑙 ∈ ran 𝐺 𝑘𝑄𝑘𝑄))
5652, 55mpd 15 . . . . . . 7 ((𝜑𝑘 ∈ ran 𝑌) → 𝑘𝑄)
5756ex 399 . . . . . 6 (𝜑 → (𝑘 ∈ ran 𝑌𝑘𝑄))
5857ssrdv 3798 . . . . 5 (𝜑 → ran 𝑌𝑄)
5918, 58syl5ss 3803 . . . 4 (𝜑 → ran (𝑌𝐹) ⊆ 𝑄)
60 df-f 6099 . . . 4 ((𝑌𝐹):(1...𝑀)⟶𝑄 ↔ ((𝑌𝐹) Fn (1...𝑀) ∧ ran (𝑌𝐹) ⊆ 𝑄))
6117, 59, 60sylanbrc 574 . . 3 (𝜑 → (𝑌𝐹):(1...𝑀)⟶𝑄)
62 stoweidlem27.9 . . . 4 𝑡𝜑
63 nfv 2005 . . . . . . 7 𝑤 𝑡 ∈ (𝑇𝑈)
6422, 63nfan 1990 . . . . . 6 𝑤(𝜑𝑡 ∈ (𝑇𝑈))
65 nfv 2005 . . . . . 6 𝑤𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)
66 stoweidlem27.8 . . . . . . . 8 (𝜑 → (𝑇𝑈) ⊆ 𝑋)
6766sselda 3792 . . . . . . 7 ((𝜑𝑡 ∈ (𝑇𝑈)) → 𝑡 𝑋)
68 eluni 4626 . . . . . . 7 (𝑡 𝑋 ↔ ∃𝑤(𝑡𝑤𝑤𝑋))
6967, 68sylib 209 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑤(𝑡𝑤𝑤𝑋))
7023funmpt2 6134 . . . . . . . . . . . 12 Fun 𝐺
7123dmeqi 5520 . . . . . . . . . . . . . . 15 dom 𝐺 = dom (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
72 stoweidlem27.2 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑄 ∈ V)
7334rabexgf 39671 . . . . . . . . . . . . . . . . . . . 20 (𝑄 ∈ V → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
7472, 73syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
7574adantr 468 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑋) → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
7675ex 399 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑤𝑋 → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V))
7722, 76ralrimi 3141 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑤𝑋 {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
78 dmmptg 5840 . . . . . . . . . . . . . . . 16 (∀𝑤𝑋 {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V → dom (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) = 𝑋)
7977, 78syl 17 . . . . . . . . . . . . . . 15 (𝜑 → dom (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) = 𝑋)
8071, 79syl5eq 2848 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐺 = 𝑋)
8180eleq2d 2867 . . . . . . . . . . . . 13 (𝜑 → (𝑤 ∈ dom 𝐺𝑤𝑋))
8281biimpar 465 . . . . . . . . . . . 12 ((𝜑𝑤𝑋) → 𝑤 ∈ dom 𝐺)
83 fvelrn 6568 . . . . . . . . . . . 12 ((Fun 𝐺𝑤 ∈ dom 𝐺) → (𝐺𝑤) ∈ ran 𝐺)
8470, 82, 83sylancr 577 . . . . . . . . . . 11 ((𝜑𝑤𝑋) → (𝐺𝑤) ∈ ran 𝐺)
8584adantrl 698 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝑤𝑤𝑋)) → (𝐺𝑤) ∈ ran 𝐺)
8615ad2antrr 708 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → 𝐹:(1...𝑀)⟶ran 𝐺)
87 simprl 778 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → 𝑖 ∈ (1...𝑀))
88 fvco3 6490 . . . . . . . . . . . . . 14 ((𝐹:(1...𝑀)⟶ran 𝐺𝑖 ∈ (1...𝑀)) → ((𝑌𝐹)‘𝑖) = (𝑌‘(𝐹𝑖)))
8986, 87, 88syl2anc 575 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → ((𝑌𝐹)‘𝑖) = (𝑌‘(𝐹𝑖)))
90 fveq2 6402 . . . . . . . . . . . . . 14 ((𝐹𝑖) = (𝐺𝑤) → (𝑌‘(𝐹𝑖)) = (𝑌‘(𝐺𝑤)))
9190ad2antll 711 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → (𝑌‘(𝐹𝑖)) = (𝑌‘(𝐺𝑤)))
9289, 91eqtrd 2836 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → ((𝑌𝐹)‘𝑖) = (𝑌‘(𝐺𝑤)))
93 eleq1 2869 . . . . . . . . . . . . . . . . 17 (𝑙 = (𝐺𝑤) → (𝑙 ∈ ran 𝐺 ↔ (𝐺𝑤) ∈ ran 𝐺))
9493anbi2d 616 . . . . . . . . . . . . . . . 16 (𝑙 = (𝐺𝑤) → ((𝜑𝑙 ∈ ran 𝐺) ↔ (𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺)))
95 eleq2 2870 . . . . . . . . . . . . . . . . 17 (𝑙 = (𝐺𝑤) → ((𝑌𝑙) ∈ 𝑙 ↔ (𝑌𝑙) ∈ (𝐺𝑤)))
96 fveq2 6402 . . . . . . . . . . . . . . . . . 18 (𝑙 = (𝐺𝑤) → (𝑌𝑙) = (𝑌‘(𝐺𝑤)))
9796eleq1d 2866 . . . . . . . . . . . . . . . . 17 (𝑙 = (𝐺𝑤) → ((𝑌𝑙) ∈ (𝐺𝑤) ↔ (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤)))
9895, 97bitrd 270 . . . . . . . . . . . . . . . 16 (𝑙 = (𝐺𝑤) → ((𝑌𝑙) ∈ 𝑙 ↔ (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤)))
9994, 98imbi12d 335 . . . . . . . . . . . . . . 15 (𝑙 = (𝐺𝑤) → (((𝜑𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑙) ↔ ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤))))
10099, 29vtoclg 3455 . . . . . . . . . . . . . 14 ((𝐺𝑤) ∈ ran 𝐺 → ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤)))
101100anabsi7 653 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤))
102101adantr 468 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤))
10392, 102eqeltrd 2881 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤))
104 f1ofo 6354 . . . . . . . . . . . . . . 15 (𝐹:(1...𝑀)–1-1-onto→ran 𝐺𝐹:(1...𝑀)–onto→ran 𝐺)
105 forn 6328 . . . . . . . . . . . . . . 15 (𝐹:(1...𝑀)–onto→ran 𝐺 → ran 𝐹 = ran 𝐺)
1065, 104, 1053syl 18 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 = ran 𝐺)
107106eleq2d 2867 . . . . . . . . . . . . 13 (𝜑 → ((𝐺𝑤) ∈ ran 𝐹 ↔ (𝐺𝑤) ∈ ran 𝐺))
108107biimpar 465 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → (𝐺𝑤) ∈ ran 𝐹)
1097adantr 468 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → 𝐹 Fn (1...𝑀))
110 fvelrnb 6458 . . . . . . . . . . . . 13 (𝐹 Fn (1...𝑀) → ((𝐺𝑤) ∈ ran 𝐹 ↔ ∃𝑖 ∈ (1...𝑀)(𝐹𝑖) = (𝐺𝑤)))
111109, 110syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → ((𝐺𝑤) ∈ ran 𝐹 ↔ ∃𝑖 ∈ (1...𝑀)(𝐹𝑖) = (𝐺𝑤)))
112108, 111mpbid 223 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → ∃𝑖 ∈ (1...𝑀)(𝐹𝑖) = (𝐺𝑤))
113103, 112reximddv 3201 . . . . . . . . . 10 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → ∃𝑖 ∈ (1...𝑀)((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤))
11485, 113syldan 581 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑤𝑤𝑋)) → ∃𝑖 ∈ (1...𝑀)((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤))
115 simplrl 786 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → 𝑡𝑤)
116 simpr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤𝑋) → 𝑤𝑋)
11723fvmpt2 6506 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝑋 ∧ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V) → (𝐺𝑤) = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
118116, 75, 117syl2anc 575 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑋) → (𝐺𝑤) = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
119118eleq2d 2867 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑋) → (((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤) ↔ ((𝑌𝐹)‘𝑖) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}))
120119biimpa 464 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝑋) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → ((𝑌𝐹)‘𝑖) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
121120adantlrl 702 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → ((𝑌𝐹)‘𝑖) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
122 nfcv 2944 . . . . . . . . . . . . . . . . 17 ((𝑌𝐹)‘𝑖)
123 nfv 2005 . . . . . . . . . . . . . . . . 17 𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)}
124 fveq1 6401 . . . . . . . . . . . . . . . . . . . 20 ( = ((𝑌𝐹)‘𝑖) → (𝑡) = (((𝑌𝐹)‘𝑖)‘𝑡))
125124breq2d 4849 . . . . . . . . . . . . . . . . . . 19 ( = ((𝑌𝐹)‘𝑖) → (0 < (𝑡) ↔ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
126125rabbidv 3375 . . . . . . . . . . . . . . . . . 18 ( = ((𝑌𝐹)‘𝑖) → {𝑡𝑇 ∣ 0 < (𝑡)} = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)})
127126eqeq2d 2812 . . . . . . . . . . . . . . . . 17 ( = ((𝑌𝐹)‘𝑖) → (𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)} ↔ 𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)}))
128122, 34, 123, 127elrabf 3551 . . . . . . . . . . . . . . . 16 (((𝑌𝐹)‘𝑖) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ (((𝑌𝐹)‘𝑖) ∈ 𝑄𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)}))
129121, 128sylib 209 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → (((𝑌𝐹)‘𝑖) ∈ 𝑄𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)}))
130129simprd 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → 𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)})
131115, 130eleqtrd 2883 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → 𝑡 ∈ {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)})
132 rabid 3300 . . . . . . . . . . . . 13 (𝑡 ∈ {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)} ↔ (𝑡𝑇 ∧ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
133131, 132sylib 209 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → (𝑡𝑇 ∧ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
134133simprd 485 . . . . . . . . . . 11 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → 0 < (((𝑌𝐹)‘𝑖)‘𝑡))
135134ex 399 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝑤𝑤𝑋)) → (((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤) → 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
136135reximdv 3199 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑤𝑤𝑋)) → (∃𝑖 ∈ (1...𝑀)((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
137114, 136mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑡𝑤𝑤𝑋)) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡))
138137ex 399 . . . . . . 7 (𝜑 → ((𝑡𝑤𝑤𝑋) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
139138adantr 468 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → ((𝑡𝑤𝑤𝑋) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
14064, 65, 69, 139exlimimdd 2255 . . . . 5 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡))
141140ex 399 . . . 4 (𝜑 → (𝑡 ∈ (𝑇𝑈) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
14262, 141ralrimi 3141 . . 3 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡))
14313, 61, 142jca32 507 . 2 (𝜑 → (𝑀 ∈ ℕ ∧ ((𝑌𝐹):(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡))))
144 feq1 6231 . . . . 5 (𝑞 = (𝑌𝐹) → (𝑞:(1...𝑀)⟶𝑄 ↔ (𝑌𝐹):(1...𝑀)⟶𝑄))
145 fveq1 6401 . . . . . . . . 9 (𝑞 = (𝑌𝐹) → (𝑞𝑖) = ((𝑌𝐹)‘𝑖))
146145fveq1d 6404 . . . . . . . 8 (𝑞 = (𝑌𝐹) → ((𝑞𝑖)‘𝑡) = (((𝑌𝐹)‘𝑖)‘𝑡))
147146breq2d 4849 . . . . . . 7 (𝑞 = (𝑌𝐹) → (0 < ((𝑞𝑖)‘𝑡) ↔ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
148147rexbidv 3236 . . . . . 6 (𝑞 = (𝑌𝐹) → (∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡) ↔ ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
149148ralbidv 3170 . . . . 5 (𝑞 = (𝑌𝐹) → (∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
150144, 149anbi12d 618 . . . 4 (𝑞 = (𝑌𝐹) → ((𝑞:(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡)) ↔ ((𝑌𝐹):(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡))))
151150anbi2d 616 . . 3 (𝑞 = (𝑌𝐹) → ((𝑀 ∈ ℕ ∧ (𝑞:(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡))) ↔ (𝑀 ∈ ℕ ∧ ((𝑌𝐹):(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))))
152151spcegv 3483 . 2 ((𝑌𝐹) ∈ V → ((𝑀 ∈ ℕ ∧ ((𝑌𝐹):(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡))) → ∃𝑞(𝑀 ∈ ℕ ∧ (𝑞:(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡)))))
15312, 143, 152sylc 65 1 (𝜑 → ∃𝑞(𝑀 ∈ ℕ ∧ (𝑞:(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wex 1859  wnf 1863  wcel 2155  wnfc 2931  wral 3092  wrex 3093  {crab 3096  Vcvv 3387  cdif 3760  wss 3763   cuni 4623   class class class wbr 4837  cmpt 4916  dom cdm 5305  ran crn 5306  ccom 5309  Fun wfun 6089   Fn wfn 6090  wf 6091  ontowfo 6093  1-1-ontowf1o 6094  cfv 6095  (class class class)co 6868  0cc0 10215  1c1 10216   < clt 10353  cn 11299  ...cfz 12543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871
This theorem is referenced by:  stoweidlem35  40725
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