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Theorem stoweidlem27 42622
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_(t0) = 0, and p > 0 on T - U. Here (𝑞𝑖) is used to represent p_(ti) 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 6973 . . . 4 ((𝑌 Fn ran 𝐺 ∧ ran 𝐺 ∈ V) → 𝑌 ∈ V)
41, 2, 3syl2anc 587 . . 3 (𝜑𝑌 ∈ V)
5 stoweidlem27.7 . . . . 5 (𝜑𝐹:(1...𝑀)–1-1-onto→ran 𝐺)
6 f1ofn 6609 . . . . 5 (𝐹:(1...𝑀)–1-1-onto→ran 𝐺𝐹 Fn (1...𝑀))
75, 6syl 17 . . . 4 (𝜑𝐹 Fn (1...𝑀))
8 ovex 7184 . . . 4 (1...𝑀) ∈ V
9 fnex 6973 . . . 4 ((𝐹 Fn (1...𝑀) ∧ (1...𝑀) ∈ V) → 𝐹 ∈ V)
107, 8, 9sylancl 589 . . 3 (𝜑𝐹 ∈ V)
11 coexg 7631 . . 3 ((𝑌 ∈ V ∧ 𝐹 ∈ V) → (𝑌𝐹) ∈ V)
124, 10, 11syl2anc 587 . 2 (𝜑 → (𝑌𝐹) ∈ V)
13 stoweidlem27.3 . . 3 (𝜑𝑀 ∈ ℕ)
14 f1of 6608 . . . . . 6 (𝐹:(1...𝑀)–1-1-onto→ran 𝐺𝐹:(1...𝑀)⟶ran 𝐺)
155, 14syl 17 . . . . 5 (𝜑𝐹:(1...𝑀)⟶ran 𝐺)
16 fnfco 6535 . . . . 5 ((𝑌 Fn ran 𝐺𝐹:(1...𝑀)⟶ran 𝐺) → (𝑌𝐹) Fn (1...𝑀))
171, 15, 16syl2anc 587 . . . 4 (𝜑 → (𝑌𝐹) Fn (1...𝑀))
18 rncoss 5832 . . . . 5 ran (𝑌𝐹) ⊆ ran 𝑌
19 fvelrnb 6719 . . . . . . . . . . 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 480 . . . . . . . . 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 5151 . . . . . . . . . . . . . . . . 17 𝑤(𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
2523, 24nfcxfr 2980 . . . . . . . . . . . . . . . 16 𝑤𝐺
2625nfrn 5812 . . . . . . . . . . . . . . 15 𝑤ran 𝐺
2726nfcri 2969 . . . . . . . . . . . . . 14 𝑤 𝑙 ∈ ran 𝐺
2822, 27nfan 1901 . . . . . . . . . . . . 13 𝑤(𝜑𝑙 ∈ ran 𝐺)
29 stoweidlem27.6 . . . . . . . . . . . . . . . . 17 ((𝜑𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑙)
3029ad2antrr 725 . . . . . . . . . . . . . . . 16 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → (𝑌𝑙) ∈ 𝑙)
31 simpr 488 . . . . . . . . . . . . . . . 16 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
3230, 31eleqtrd 2918 . . . . . . . . . . . . . . 15 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → (𝑌𝑙) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
33 nfcv 2982 . . . . . . . . . . . . . . . 16 (𝑌𝑙)
34 stoweidlem27.11 . . . . . . . . . . . . . . . 16 𝑄
35 nfv 1916 . . . . . . . . . . . . . . . 16 𝑤 = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)}
36 fveq1 6662 . . . . . . . . . . . . . . . . . . 19 ( = (𝑌𝑙) → (𝑡) = ((𝑌𝑙)‘𝑡))
3736breq2d 5065 . . . . . . . . . . . . . . . . . 18 ( = (𝑌𝑙) → (0 < (𝑡) ↔ 0 < ((𝑌𝑙)‘𝑡)))
3837rabbidv 3465 . . . . . . . . . . . . . . . . 17 ( = (𝑌𝑙) → {𝑡𝑇 ∣ 0 < (𝑡)} = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)})
3938eqeq2d 2835 . . . . . . . . . . . . . . . 16 ( = (𝑌𝑙) → (𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)} ↔ 𝑤 = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)}))
4033, 34, 35, 39elrabf 3662 . . . . . . . . . . . . . . 15 ((𝑌𝑙) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ ((𝑌𝑙) ∈ 𝑄𝑤 = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)}))
4132, 40sylib 221 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ((𝑌𝑙) ∈ 𝑄𝑤 = {𝑡𝑇 ∣ 0 < ((𝑌𝑙)‘𝑡)}))
4241simpld 498 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → (𝑌𝑙) ∈ 𝑄)
43 simpr 488 . . . . . . . . . . . . . 14 ((𝜑𝑙 ∈ ran 𝐺) → 𝑙 ∈ ran 𝐺)
4423elrnmpt 5816 . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . 13 ((𝜑𝑙 ∈ ran 𝐺) → ∃𝑤𝑋 𝑙 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
4728, 42, 46r19.29af 3322 . . . . . . . . . . . 12 ((𝜑𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑄)
4847adantlr 714 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ran 𝑌) ∧ 𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑄)
49 eleq1 2903 . . . . . . . . . . 11 ((𝑌𝑙) = 𝑘 → ((𝑌𝑙) ∈ 𝑄𝑘𝑄))
5048, 49syl5ibcom 248 . . . . . . . . . 10 (((𝜑𝑘 ∈ ran 𝑌) ∧ 𝑙 ∈ ran 𝐺) → ((𝑌𝑙) = 𝑘𝑘𝑄))
5150reximdva 3266 . . . . . . . . 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 3275 . . . . . . . 8 ((𝜑𝑘 ∈ ran 𝑌) → (∃𝑙 ∈ ran 𝐺 𝑘𝑄𝑘𝑄))
5652, 55mpd 15 . . . . . . 7 ((𝜑𝑘 ∈ ran 𝑌) → 𝑘𝑄)
5756ex 416 . . . . . 6 (𝜑 → (𝑘 ∈ ran 𝑌𝑘𝑄))
5857ssrdv 3959 . . . . 5 (𝜑 → ran 𝑌𝑄)
5918, 58sstrid 3964 . . . 4 (𝜑 → ran (𝑌𝐹) ⊆ 𝑄)
60 df-f 6349 . . . 4 ((𝑌𝐹):(1...𝑀)⟶𝑄 ↔ ((𝑌𝐹) Fn (1...𝑀) ∧ ran (𝑌𝐹) ⊆ 𝑄))
6117, 59, 60sylanbrc 586 . . 3 (𝜑 → (𝑌𝐹):(1...𝑀)⟶𝑄)
62 stoweidlem27.9 . . . 4 𝑡𝜑
63 nfv 1916 . . . . . . 7 𝑤 𝑡 ∈ (𝑇𝑈)
6422, 63nfan 1901 . . . . . 6 𝑤(𝜑𝑡 ∈ (𝑇𝑈))
65 nfv 1916 . . . . . 6 𝑤𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)
66 stoweidlem27.8 . . . . . . . 8 (𝜑 → (𝑇𝑈) ⊆ 𝑋)
6766sselda 3953 . . . . . . 7 ((𝜑𝑡 ∈ (𝑇𝑈)) → 𝑡 𝑋)
68 eluni 4827 . . . . . . 7 (𝑡 𝑋 ↔ ∃𝑤(𝑡𝑤𝑤𝑋))
6967, 68sylib 221 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑤(𝑡𝑤𝑤𝑋))
7023funmpt2 6384 . . . . . . . . . . . 12 Fun 𝐺
7123dmeqi 5761 . . . . . . . . . . . . . . 15 dom 𝐺 = dom (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
72 stoweidlem27.2 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑄 ∈ V)
7334rabexgf 41601 . . . . . . . . . . . . . . . . . . . 20 (𝑄 ∈ V → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
7472, 73syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
7574adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑋) → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
7675ex 416 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑤𝑋 → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V))
7722, 76ralrimi 3210 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑤𝑋 {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
78 dmmptg 6085 . . . . . . . . . . . . . . . 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 2871 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐺 = 𝑋)
8180eleq2d 2901 . . . . . . . . . . . . 13 (𝜑 → (𝑤 ∈ dom 𝐺𝑤𝑋))
8281biimpar 481 . . . . . . . . . . . 12 ((𝜑𝑤𝑋) → 𝑤 ∈ dom 𝐺)
83 fvelrn 6837 . . . . . . . . . . . 12 ((Fun 𝐺𝑤 ∈ dom 𝐺) → (𝐺𝑤) ∈ ran 𝐺)
8470, 82, 83sylancr 590 . . . . . . . . . . 11 ((𝜑𝑤𝑋) → (𝐺𝑤) ∈ ran 𝐺)
8584adantrl 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝑤𝑤𝑋)) → (𝐺𝑤) ∈ ran 𝐺)
8615ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → 𝐹:(1...𝑀)⟶ran 𝐺)
87 simprl 770 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → 𝑖 ∈ (1...𝑀))
88 fvco3 6753 . . . . . . . . . . . . . 14 ((𝐹:(1...𝑀)⟶ran 𝐺𝑖 ∈ (1...𝑀)) → ((𝑌𝐹)‘𝑖) = (𝑌‘(𝐹𝑖)))
8986, 87, 88syl2anc 587 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → ((𝑌𝐹)‘𝑖) = (𝑌‘(𝐹𝑖)))
90 fveq2 6663 . . . . . . . . . . . . . 14 ((𝐹𝑖) = (𝐺𝑤) → (𝑌‘(𝐹𝑖)) = (𝑌‘(𝐺𝑤)))
9190ad2antll 728 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → (𝑌‘(𝐹𝑖)) = (𝑌‘(𝐺𝑤)))
9289, 91eqtrd 2859 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → ((𝑌𝐹)‘𝑖) = (𝑌‘(𝐺𝑤)))
93 eleq1 2903 . . . . . . . . . . . . . . . . 17 (𝑙 = (𝐺𝑤) → (𝑙 ∈ ran 𝐺 ↔ (𝐺𝑤) ∈ ran 𝐺))
9493anbi2d 631 . . . . . . . . . . . . . . . 16 (𝑙 = (𝐺𝑤) → ((𝜑𝑙 ∈ ran 𝐺) ↔ (𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺)))
95 eleq2 2904 . . . . . . . . . . . . . . . . 17 (𝑙 = (𝐺𝑤) → ((𝑌𝑙) ∈ 𝑙 ↔ (𝑌𝑙) ∈ (𝐺𝑤)))
96 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑙 = (𝐺𝑤) → (𝑌𝑙) = (𝑌‘(𝐺𝑤)))
9796eleq1d 2900 . . . . . . . . . . . . . . . . 17 (𝑙 = (𝐺𝑤) → ((𝑌𝑙) ∈ (𝐺𝑤) ↔ (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤)))
9895, 97bitrd 282 . . . . . . . . . . . . . . . 16 (𝑙 = (𝐺𝑤) → ((𝑌𝑙) ∈ 𝑙 ↔ (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤)))
9994, 98imbi12d 348 . . . . . . . . . . . . . . 15 (𝑙 = (𝐺𝑤) → (((𝜑𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑙) ↔ ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤))))
10099, 29vtoclg 3553 . . . . . . . . . . . . . 14 ((𝐺𝑤) ∈ ran 𝐺 → ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤)))
101100anabsi7 670 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤))
102101adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → (𝑌‘(𝐺𝑤)) ∈ (𝐺𝑤))
10392, 102eqeltrd 2916 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (𝐹𝑖) = (𝐺𝑤))) → ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤))
104 f1ofo 6615 . . . . . . . . . . . . . . 15 (𝐹:(1...𝑀)–1-1-onto→ran 𝐺𝐹:(1...𝑀)–onto→ran 𝐺)
105 forn 6586 . . . . . . . . . . . . . . 15 (𝐹:(1...𝑀)–onto→ran 𝐺 → ran 𝐹 = ran 𝐺)
1065, 104, 1053syl 18 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 = ran 𝐺)
107106eleq2d 2901 . . . . . . . . . . . . 13 (𝜑 → ((𝐺𝑤) ∈ ran 𝐹 ↔ (𝐺𝑤) ∈ ran 𝐺))
108107biimpar 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → (𝐺𝑤) ∈ ran 𝐹)
1097adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → 𝐹 Fn (1...𝑀))
110 fvelrnb 6719 . . . . . . . . . . . . 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 235 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → ∃𝑖 ∈ (1...𝑀)(𝐹𝑖) = (𝐺𝑤))
113103, 112reximddv 3267 . . . . . . . . . 10 ((𝜑 ∧ (𝐺𝑤) ∈ ran 𝐺) → ∃𝑖 ∈ (1...𝑀)((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤))
11485, 113syldan 594 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑤𝑤𝑋)) → ∃𝑖 ∈ (1...𝑀)((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤))
115 simplrl 776 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → 𝑡𝑤)
116 simpr 488 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤𝑋) → 𝑤𝑋)
11723fvmpt2 6772 . . . . . . . . . . . . . . . . . . . 20 ((𝑤𝑋 ∧ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V) → (𝐺𝑤) = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
118116, 75, 117syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤𝑋) → (𝐺𝑤) = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
119118eleq2d 2901 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝑋) → (((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤) ↔ ((𝑌𝐹)‘𝑖) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}))
120119biimpa 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤𝑋) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → ((𝑌𝐹)‘𝑖) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
121120adantlrl 719 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → ((𝑌𝐹)‘𝑖) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
122 nfcv 2982 . . . . . . . . . . . . . . . . 17 ((𝑌𝐹)‘𝑖)
123 nfv 1916 . . . . . . . . . . . . . . . . 17 𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)}
124 fveq1 6662 . . . . . . . . . . . . . . . . . . . 20 ( = ((𝑌𝐹)‘𝑖) → (𝑡) = (((𝑌𝐹)‘𝑖)‘𝑡))
125124breq2d 5065 . . . . . . . . . . . . . . . . . . 19 ( = ((𝑌𝐹)‘𝑖) → (0 < (𝑡) ↔ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
126125rabbidv 3465 . . . . . . . . . . . . . . . . . 18 ( = ((𝑌𝐹)‘𝑖) → {𝑡𝑇 ∣ 0 < (𝑡)} = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)})
127126eqeq2d 2835 . . . . . . . . . . . . . . . . 17 ( = ((𝑌𝐹)‘𝑖) → (𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)} ↔ 𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)}))
128122, 34, 123, 127elrabf 3662 . . . . . . . . . . . . . . . 16 (((𝑌𝐹)‘𝑖) ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ (((𝑌𝐹)‘𝑖) ∈ 𝑄𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)}))
129121, 128sylib 221 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → (((𝑌𝐹)‘𝑖) ∈ 𝑄𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)}))
130129simprd 499 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → 𝑤 = {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)})
131115, 130eleqtrd 2918 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → 𝑡 ∈ {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)})
132 rabid 3369 . . . . . . . . . . . . 13 (𝑡 ∈ {𝑡𝑇 ∣ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)} ↔ (𝑡𝑇 ∧ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
133131, 132sylib 221 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → (𝑡𝑇 ∧ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
134133simprd 499 . . . . . . . . . . 11 (((𝜑 ∧ (𝑡𝑤𝑤𝑋)) ∧ ((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤)) → 0 < (((𝑌𝐹)‘𝑖)‘𝑡))
135134ex 416 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝑤𝑤𝑋)) → (((𝑌𝐹)‘𝑖) ∈ (𝐺𝑤) → 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
136135reximdv 3265 . . . . . . . . 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 416 . . . . . . 7 (𝜑 → ((𝑡𝑤𝑤𝑋) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
139138adantr 484 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → ((𝑡𝑤𝑤𝑋) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
14064, 65, 69, 139exlimimdd 2221 . . . . 5 ((𝜑𝑡 ∈ (𝑇𝑈)) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡))
141140ex 416 . . . 4 (𝜑 → (𝑡 ∈ (𝑇𝑈) → ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
14262, 141ralrimi 3210 . . 3 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡))
14313, 61, 142jca32 519 . 2 (𝜑 → (𝑀 ∈ ℕ ∧ ((𝑌𝐹):(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡))))
144 feq1 6486 . . . . 5 (𝑞 = (𝑌𝐹) → (𝑞:(1...𝑀)⟶𝑄 ↔ (𝑌𝐹):(1...𝑀)⟶𝑄))
145 fveq1 6662 . . . . . . . . 9 (𝑞 = (𝑌𝐹) → (𝑞𝑖) = ((𝑌𝐹)‘𝑖))
146145fveq1d 6665 . . . . . . . 8 (𝑞 = (𝑌𝐹) → ((𝑞𝑖)‘𝑡) = (((𝑌𝐹)‘𝑖)‘𝑡))
147146breq2d 5065 . . . . . . 7 (𝑞 = (𝑌𝐹) → (0 < ((𝑞𝑖)‘𝑡) ↔ 0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
148147rexbidv 3289 . . . . . 6 (𝑞 = (𝑌𝐹) → (∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡) ↔ ∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
149148ralbidv 3192 . . . . 5 (𝑞 = (𝑌𝐹) → (∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < (((𝑌𝐹)‘𝑖)‘𝑡)))
150144, 149anbi12d 633 . . . 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 631 . . 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 3583 . 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 209  wa 399   = wceq 1538  wex 1781  wnf 1785  wcel 2115  wnfc 2962  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  cdif 3916  wss 3919   cuni 4824   class class class wbr 5053  cmpt 5133  dom cdm 5543  ran crn 5544  ccom 5547  Fun wfun 6339   Fn wfn 6340  wf 6341  ontowfo 6343  1-1-ontowf1o 6344  cfv 6345  (class class class)co 7151  0cc0 10537  1c1 10538   < clt 10675  cn 11636  ...cfz 12896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154
This theorem is referenced by:  stoweidlem35  42630
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