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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem35 Structured version   Visualization version   GIF version
Theorem stoweidlem35 44836
Description: This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Here (π‘žβ€˜π‘–) is used to represent p_(ti) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem35.1 β„²π‘‘πœ‘
stoweidlem35.2 β„²π‘€πœ‘
stoweidlem35.3 β„²β„Žπœ‘
stoweidlem35.4 𝑄 = {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))}
stoweidlem35.5 π‘Š = {𝑀 ∈ 𝐽 ∣ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}
stoweidlem35.6 𝐺 = (𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
stoweidlem35.7 (πœ‘ β†’ 𝐴 ∈ V)
stoweidlem35.8 (πœ‘ β†’ 𝑋 ∈ Fin)
stoweidlem35.9 (πœ‘ β†’ 𝑋 βŠ† π‘Š)
stoweidlem35.10 (πœ‘ β†’ (𝑇 βˆ– π‘ˆ) βŠ† βˆͺ 𝑋)
stoweidlem35.11 (πœ‘ β†’ (𝑇 βˆ– π‘ˆ) β‰  βˆ…)
Assertion
Ref Expression
stoweidlem35 (πœ‘ β†’ βˆƒπ‘šβˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))))
Distinct variable groups:   β„Ž,𝑖,𝑑,𝑀   𝑖,π‘š,π‘ž,𝑑   𝑖,𝐺   𝑀,𝑄   𝑇,β„Ž,𝑀   π‘ˆ,π‘ž   πœ‘,𝑖,π‘š   𝐴,β„Ž,𝑑   β„Ž,𝑋,𝑖,𝑑,𝑀   𝑀,π‘š   π‘š,𝐺   𝑄,π‘ž   𝑇,π‘ž   𝑑,𝑍   𝑀,π‘ˆ
Allowed substitution hints:   πœ‘(𝑀,𝑑,β„Ž,π‘ž)   𝐴(𝑀,𝑖,π‘š,π‘ž)   𝑄(𝑑,β„Ž,𝑖,π‘š)   𝑇(𝑑,𝑖,π‘š)   π‘ˆ(𝑑,β„Ž,𝑖,π‘š)   𝐺(𝑀,𝑑,β„Ž,π‘ž)   𝐽(𝑀,𝑑,β„Ž,𝑖,π‘š,π‘ž)   π‘Š(𝑀,𝑑,β„Ž,𝑖,π‘š,π‘ž)   𝑋(π‘š,π‘ž)   𝑍(𝑀,β„Ž,𝑖,π‘š,π‘ž)
Proof of Theorem stoweidlem35
Dummy variables 𝑓 𝑔 π‘˜ 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem35.8 . . . . . . . . . 10 (πœ‘ β†’ 𝑋 ∈ Fin)
2 stoweidlem35.6 . . . . . . . . . . 11 𝐺 = (𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
32rnmptfi 43955 . . . . . . . . . 10 (𝑋 ∈ Fin β†’ ran 𝐺 ∈ Fin)
41, 3syl 17 . . . . . . . . 9 (πœ‘ β†’ ran 𝐺 ∈ Fin)
5 fnchoice 43801 . . . . . . . . . . 11 (ran 𝐺 ∈ Fin β†’ βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙)))
65adantl 482 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐺 ∈ Fin) β†’ βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙)))
7 simprl 769 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙))) β†’ 𝑔 Fn ran 𝐺)
8 stoweidlem35.2 . . . . . . . . . . . . . . . . . . . . 21 β„²π‘€πœ‘
9 nfmpt1 5256 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑀(𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
102, 9nfcxfr 2901 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑀𝐺
1110nfrn 5951 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑀ran 𝐺
1211nfcri 2890 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑀 π‘˜ ∈ ran 𝐺
138, 12nfan 1902 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑀(πœ‘ ∧ π‘˜ ∈ ran 𝐺)
14 stoweidlem35.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (πœ‘ β†’ 𝑋 βŠ† π‘Š)
1514sselda 3982 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ 𝑀 ∈ π‘Š)
16 stoweidlem35.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 π‘Š = {𝑀 ∈ 𝐽 ∣ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}
1715, 16eleqtrdi 2843 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ 𝑀 ∈ {𝑀 ∈ 𝐽 ∣ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
18 rabid 3452 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ {𝑀 ∈ 𝐽 ∣ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ↔ (𝑀 ∈ 𝐽 ∧ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}))
1917, 18sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ (𝑀 ∈ 𝐽 ∧ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}))
2019simprd 496 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)})
21 df-rex 3071 . . . . . . . . . . . . . . . . . . . . . . . . 25 (βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)} ↔ βˆƒβ„Ž(β„Ž ∈ 𝑄 ∧ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}))
2220, 21sylib 217 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ βˆƒβ„Ž(β„Ž ∈ 𝑄 ∧ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}))
23 rabid 3452 . . . . . . . . . . . . . . . . . . . . . . . . 25 (β„Ž ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ↔ (β„Ž ∈ 𝑄 ∧ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}))
2423exbii 1850 . . . . . . . . . . . . . . . . . . . . . . . 24 (βˆƒβ„Ž β„Ž ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ↔ βˆƒβ„Ž(β„Ž ∈ 𝑄 ∧ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}))
2522, 24sylibr 233 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ βˆƒβ„Ž β„Ž ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
2625adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ 𝑀 ∈ 𝑋) ∧ π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ βˆƒβ„Ž β„Ž ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
27 stoweidlem35.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 β„²β„Žπœ‘
28 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . . 25 β„²β„Ž 𝑀 ∈ 𝑋
2927, 28nfan 1902 . . . . . . . . . . . . . . . . . . . . . . . 24 β„²β„Ž(πœ‘ ∧ 𝑀 ∈ 𝑋)
30 nfrab1 3451 . . . . . . . . . . . . . . . . . . . . . . . . 25 β„²β„Ž{β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}
3130nfeq2 2920 . . . . . . . . . . . . . . . . . . . . . . . 24 β„²β„Ž π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}
3229, 31nfan 1902 . . . . . . . . . . . . . . . . . . . . . . 23 β„²β„Ž((πœ‘ ∧ 𝑀 ∈ 𝑋) ∧ π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
33 eleq2 2822 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} β†’ (β„Ž ∈ π‘˜ ↔ β„Ž ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}))
3433biimprd 247 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} β†’ (β„Ž ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} β†’ β„Ž ∈ π‘˜))
3534adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ 𝑀 ∈ 𝑋) ∧ π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ (β„Ž ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} β†’ β„Ž ∈ π‘˜))
3632, 35eximd 2209 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ 𝑀 ∈ 𝑋) ∧ π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ (βˆƒβ„Ž β„Ž ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} β†’ βˆƒβ„Ž β„Ž ∈ π‘˜))
3726, 36mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ 𝑀 ∈ 𝑋) ∧ π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ βˆƒβ„Ž β„Ž ∈ π‘˜)
3837adantllr 717 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘˜ ∈ ran 𝐺) ∧ 𝑀 ∈ 𝑋) ∧ π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ βˆƒβ„Ž β„Ž ∈ π‘˜)
392elrnmpt 5955 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ ∈ ran 𝐺 β†’ (π‘˜ ∈ ran 𝐺 ↔ βˆƒπ‘€ ∈ 𝑋 π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}))
4039ibi 266 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ ∈ ran 𝐺 β†’ βˆƒπ‘€ ∈ 𝑋 π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
4140adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘˜ ∈ ran 𝐺) β†’ βˆƒπ‘€ ∈ 𝑋 π‘˜ = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
4213, 38, 41r19.29af 3265 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘˜ ∈ ran 𝐺) β†’ βˆƒβ„Ž β„Ž ∈ π‘˜)
43 n0 4346 . . . . . . . . . . . . . . . . . . 19 (π‘˜ β‰  βˆ… ↔ βˆƒβ„Ž β„Ž ∈ π‘˜)
4442, 43sylibr 233 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘˜ ∈ ran 𝐺) β†’ π‘˜ β‰  βˆ…)
4544adantlr 713 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙))) ∧ π‘˜ ∈ ran 𝐺) β†’ π‘˜ β‰  βˆ…)
46 simplrr 776 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙))) ∧ π‘˜ ∈ ran 𝐺) β†’ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙))
47 neeq1 3003 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = π‘˜ β†’ (𝑙 β‰  βˆ… ↔ π‘˜ β‰  βˆ…))
48 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = π‘˜ β†’ (π‘”β€˜π‘™) = (π‘”β€˜π‘˜))
4948eleq1d 2818 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = π‘˜ β†’ ((π‘”β€˜π‘™) ∈ 𝑙 ↔ (π‘”β€˜π‘˜) ∈ 𝑙))
50 eleq2 2822 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = π‘˜ β†’ ((π‘”β€˜π‘˜) ∈ 𝑙 ↔ (π‘”β€˜π‘˜) ∈ π‘˜))
5149, 50bitrd 278 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = π‘˜ β†’ ((π‘”β€˜π‘™) ∈ 𝑙 ↔ (π‘”β€˜π‘˜) ∈ π‘˜))
5247, 51imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑙 = π‘˜ β†’ ((𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙) ↔ (π‘˜ β‰  βˆ… β†’ (π‘”β€˜π‘˜) ∈ π‘˜)))
5352rspccva 3611 . . . . . . . . . . . . . . . . . 18 ((βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙) ∧ π‘˜ ∈ ran 𝐺) β†’ (π‘˜ β‰  βˆ… β†’ (π‘”β€˜π‘˜) ∈ π‘˜))
5446, 53sylancom 588 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙))) ∧ π‘˜ ∈ ran 𝐺) β†’ (π‘˜ β‰  βˆ… β†’ (π‘”β€˜π‘˜) ∈ π‘˜))
5545, 54mpd 15 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙))) ∧ π‘˜ ∈ ran 𝐺) β†’ (π‘”β€˜π‘˜) ∈ π‘˜)
5655ralrimiva 3146 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙))) β†’ βˆ€π‘˜ ∈ ran 𝐺(π‘”β€˜π‘˜) ∈ π‘˜)
57 fveq2 6891 . . . . . . . . . . . . . . . . . 18 (π‘˜ = 𝑙 β†’ (π‘”β€˜π‘˜) = (π‘”β€˜π‘™))
5857eleq1d 2818 . . . . . . . . . . . . . . . . 17 (π‘˜ = 𝑙 β†’ ((π‘”β€˜π‘˜) ∈ π‘˜ ↔ (π‘”β€˜π‘™) ∈ π‘˜))
59 eleq2 2822 . . . . . . . . . . . . . . . . 17 (π‘˜ = 𝑙 β†’ ((π‘”β€˜π‘™) ∈ π‘˜ ↔ (π‘”β€˜π‘™) ∈ 𝑙))
6058, 59bitrd 278 . . . . . . . . . . . . . . . 16 (π‘˜ = 𝑙 β†’ ((π‘”β€˜π‘˜) ∈ π‘˜ ↔ (π‘”β€˜π‘™) ∈ 𝑙))
6160cbvralvw 3234 . . . . . . . . . . . . . . 15 (βˆ€π‘˜ ∈ ran 𝐺(π‘”β€˜π‘˜) ∈ π‘˜ ↔ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙)
6256, 61sylib 217 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙))) β†’ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙)
637, 62jca 512 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙))) β†’ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙))
6463ex 413 . . . . . . . . . . . 12 (πœ‘ β†’ ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙)) β†’ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙)))
6564adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ ran 𝐺 ∈ Fin) β†’ ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙)) β†’ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙)))
6665eximdv 1920 . . . . . . . . . 10 ((πœ‘ ∧ ran 𝐺 ∈ Fin) β†’ (βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(𝑙 β‰  βˆ… β†’ (π‘”β€˜π‘™) ∈ 𝑙)) β†’ βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙)))
676, 66mpd 15 . . . . . . . . 9 ((πœ‘ ∧ ran 𝐺 ∈ Fin) β†’ βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙))
684, 67mpdan 685 . . . . . . . 8 (πœ‘ β†’ βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙))
6968ralrimivw 3150 . . . . . . 7 (πœ‘ β†’ βˆ€π‘š ∈ β„• βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙))
70 stoweidlem35.10 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝑇 βˆ– π‘ˆ) βŠ† βˆͺ 𝑋)
71 stoweidlem35.11 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝑇 βˆ– π‘ˆ) β‰  βˆ…)
72 ssn0 4400 . . . . . . . . . . . . 13 (((𝑇 βˆ– π‘ˆ) βŠ† βˆͺ 𝑋 ∧ (𝑇 βˆ– π‘ˆ) β‰  βˆ…) β†’ βˆͺ 𝑋 β‰  βˆ…)
7370, 71, 72syl2anc 584 . . . . . . . . . . . 12 (πœ‘ β†’ βˆͺ 𝑋 β‰  βˆ…)
7473neneqd 2945 . . . . . . . . . . 11 (πœ‘ β†’ Β¬ βˆͺ 𝑋 = βˆ…)
75 unieq 4919 . . . . . . . . . . . 12 (𝑋 = βˆ… β†’ βˆͺ 𝑋 = βˆͺ βˆ…)
76 uni0 4939 . . . . . . . . . . . 12 βˆͺ βˆ… = βˆ…
7775, 76eqtrdi 2788 . . . . . . . . . . 11 (𝑋 = βˆ… β†’ βˆͺ 𝑋 = βˆ…)
7874, 77nsyl 140 . . . . . . . . . 10 (πœ‘ β†’ Β¬ 𝑋 = βˆ…)
79 dm0rn0 5924 . . . . . . . . . . 11 (dom 𝐺 = βˆ… ↔ ran 𝐺 = βˆ…)
80 stoweidlem35.4 . . . . . . . . . . . . . . . . . 18 𝑄 = {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))}
81 stoweidlem35.7 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝐴 ∈ V)
8280, 81rabexd 5333 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑄 ∈ V)
83 nfrab1 3451 . . . . . . . . . . . . . . . . . . 19 β„²β„Ž{β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))}
8480, 83nfcxfr 2901 . . . . . . . . . . . . . . . . . 18 β„²β„Žπ‘„
8584rabexgf 43796 . . . . . . . . . . . . . . . . 17 (𝑄 ∈ V β†’ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V)
8682, 85syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V)
8786adantr 481 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V)
888, 87, 2fmptdf 7118 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐺:π‘‹βŸΆV)
89 dffn2 6719 . . . . . . . . . . . . . 14 (𝐺 Fn 𝑋 ↔ 𝐺:π‘‹βŸΆV)
9088, 89sylibr 233 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐺 Fn 𝑋)
9190fndmd 6654 . . . . . . . . . . . 12 (πœ‘ β†’ dom 𝐺 = 𝑋)
9291eqeq1d 2734 . . . . . . . . . . 11 (πœ‘ β†’ (dom 𝐺 = βˆ… ↔ 𝑋 = βˆ…))
9379, 92bitr3id 284 . . . . . . . . . 10 (πœ‘ β†’ (ran 𝐺 = βˆ… ↔ 𝑋 = βˆ…))
9478, 93mtbird 324 . . . . . . . . 9 (πœ‘ β†’ Β¬ ran 𝐺 = βˆ…)
95 fz1f1o 15658 . . . . . . . . . . 11 (ran 𝐺 ∈ Fin β†’ (ran 𝐺 = βˆ… ∨ ((β™―β€˜ran 𝐺) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜ran 𝐺))–1-1-ontoβ†’ran 𝐺)))
964, 95syl 17 . . . . . . . . . 10 (πœ‘ β†’ (ran 𝐺 = βˆ… ∨ ((β™―β€˜ran 𝐺) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜ran 𝐺))–1-1-ontoβ†’ran 𝐺)))
9796ord 862 . . . . . . . . 9 (πœ‘ β†’ (Β¬ ran 𝐺 = βˆ… β†’ ((β™―β€˜ran 𝐺) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜ran 𝐺))–1-1-ontoβ†’ran 𝐺)))
9894, 97mpd 15 . . . . . . . 8 (πœ‘ β†’ ((β™―β€˜ran 𝐺) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜ran 𝐺))–1-1-ontoβ†’ran 𝐺))
99 oveq2 7419 . . . . . . . . . . 11 (π‘š = (β™―β€˜ran 𝐺) β†’ (1...π‘š) = (1...(β™―β€˜ran 𝐺)))
10099f1oeq2d 6829 . . . . . . . . . 10 (π‘š = (β™―β€˜ran 𝐺) β†’ (𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺 ↔ 𝑓:(1...(β™―β€˜ran 𝐺))–1-1-ontoβ†’ran 𝐺))
101100exbidv 1924 . . . . . . . . 9 (π‘š = (β™―β€˜ran 𝐺) β†’ (βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺 ↔ βˆƒπ‘“ 𝑓:(1...(β™―β€˜ran 𝐺))–1-1-ontoβ†’ran 𝐺))
102101rspcev 3612 . . . . . . . 8 (((β™―β€˜ran 𝐺) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜ran 𝐺))–1-1-ontoβ†’ran 𝐺) β†’ βˆƒπ‘š ∈ β„• βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)
10398, 102syl 17 . . . . . . 7 (πœ‘ β†’ βˆƒπ‘š ∈ β„• βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)
104 r19.29 3114 . . . . . . 7 ((βˆ€π‘š ∈ β„• βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ βˆƒπ‘š ∈ β„• βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺) β†’ βˆƒπ‘š ∈ β„• (βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))
10569, 103, 104syl2anc 584 . . . . . 6 (πœ‘ β†’ βˆƒπ‘š ∈ β„• (βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))
106 exdistrv 1959 . . . . . . . . 9 (βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺) ↔ (βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))
107106biimpri 227 . . . . . . . 8 ((βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺) β†’ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))
108107a1i 11 . . . . . . 7 (πœ‘ β†’ ((βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺) β†’ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
109108reximdv 3170 . . . . . 6 (πœ‘ β†’ (βˆƒπ‘š ∈ β„• (βˆƒπ‘”(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ βˆƒπ‘“ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺) β†’ βˆƒπ‘š ∈ β„• βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
110105, 109mpd 15 . . . . 5 (πœ‘ β†’ βˆƒπ‘š ∈ β„• βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))
111 df-rex 3071 . . . . 5 (βˆƒπ‘š ∈ β„• βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺) ↔ βˆƒπ‘š(π‘š ∈ β„• ∧ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
112110, 111sylib 217 . . . 4 (πœ‘ β†’ βˆƒπ‘š(π‘š ∈ β„• ∧ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
113 ax-5 1913 . . . . . . . . 9 (π‘š ∈ β„• β†’ βˆ€π‘” π‘š ∈ β„•)
114 19.29 1876 . . . . . . . . 9 ((βˆ€π‘” π‘š ∈ β„• ∧ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘”(π‘š ∈ β„• ∧ βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
115113, 114sylan 580 . . . . . . . 8 ((π‘š ∈ β„• ∧ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘”(π‘š ∈ β„• ∧ βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
116 ax-5 1913 . . . . . . . . . 10 (π‘š ∈ β„• β†’ βˆ€π‘“ π‘š ∈ β„•)
117 19.29 1876 . . . . . . . . . 10 ((βˆ€π‘“ π‘š ∈ β„• ∧ βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘“(π‘š ∈ β„• ∧ ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
118116, 117sylan 580 . . . . . . . . 9 ((π‘š ∈ β„• ∧ βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘“(π‘š ∈ β„• ∧ ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
119118eximi 1837 . . . . . . . 8 (βˆƒπ‘”(π‘š ∈ β„• ∧ βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
120115, 119syl 17 . . . . . . 7 ((π‘š ∈ β„• ∧ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
121 df-3an 1089 . . . . . . . . 9 ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺) ↔ ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))
122121anbi2i 623 . . . . . . . 8 ((π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) ↔ (π‘š ∈ β„• ∧ ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
1231222exbii 1851 . . . . . . 7 (βˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) ↔ βˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ ((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
124120, 123sylibr 233 . . . . . 6 ((π‘š ∈ β„• ∧ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
125124a1i 11 . . . . 5 (πœ‘ β†’ ((π‘š ∈ β„• ∧ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))))
126125eximdv 1920 . . . 4 (πœ‘ β†’ (βˆƒπ‘š(π‘š ∈ β„• ∧ βˆƒπ‘”βˆƒπ‘“((𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙) ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘šβˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))))
127112, 126mpd 15 . . 3 (πœ‘ β†’ βˆƒπ‘šβˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
12882adantr 481 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))) β†’ 𝑄 ∈ V)
129 simprl 769 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))) β†’ π‘š ∈ β„•)
130 simprr1 1221 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))) β†’ 𝑔 Fn ran 𝐺)
131 elex 3492 . . . . . . . . 9 (ran 𝐺 ∈ Fin β†’ ran 𝐺 ∈ V)
1324, 131syl 17 . . . . . . . 8 (πœ‘ β†’ ran 𝐺 ∈ V)
133132adantr 481 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))) β†’ ran 𝐺 ∈ V)
134 simprr2 1222 . . . . . . . 8 ((πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))) β†’ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙)
13551rspccva 3611 . . . . . . . 8 ((βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ π‘˜ ∈ ran 𝐺) β†’ (π‘”β€˜π‘˜) ∈ π‘˜)
136134, 135sylan 580 . . . . . . 7 (((πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))) ∧ π‘˜ ∈ ran 𝐺) β†’ (π‘”β€˜π‘˜) ∈ π‘˜)
137 simprr3 1223 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))) β†’ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)
13870adantr 481 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))) β†’ (𝑇 βˆ– π‘ˆ) βŠ† βˆͺ 𝑋)
139 stoweidlem35.1 . . . . . . . 8 β„²π‘‘πœ‘
140 nfv 1917 . . . . . . . . 9 Ⅎ𝑑 π‘š ∈ β„•
141 nfcv 2903 . . . . . . . . . . 11 Ⅎ𝑑𝑔
142 nfcv 2903 . . . . . . . . . . . . . 14 Ⅎ𝑑𝑋
143 nfrab1 3451 . . . . . . . . . . . . . . . 16 Ⅎ𝑑{𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}
144143nfeq2 2920 . . . . . . . . . . . . . . 15 Ⅎ𝑑 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}
145 nfv 1917 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑑(β„Žβ€˜π‘) = 0
146 nfra1 3281 . . . . . . . . . . . . . . . . . 18 β„²π‘‘βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1)
147145, 146nfan 1902 . . . . . . . . . . . . . . . . 17 Ⅎ𝑑((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))
148 nfcv 2903 . . . . . . . . . . . . . . . . 17 Ⅎ𝑑𝐴
149147, 148nfrabw 3468 . . . . . . . . . . . . . . . 16 Ⅎ𝑑{β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))}
15080, 149nfcxfr 2901 . . . . . . . . . . . . . . 15 Ⅎ𝑑𝑄
151144, 150nfrabw 3468 . . . . . . . . . . . . . 14 Ⅎ𝑑{β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}
152142, 151nfmpt 5255 . . . . . . . . . . . . 13 Ⅎ𝑑(𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
1532, 152nfcxfr 2901 . . . . . . . . . . . 12 Ⅎ𝑑𝐺
154153nfrn 5951 . . . . . . . . . . 11 Ⅎ𝑑ran 𝐺
155141, 154nffn 6648 . . . . . . . . . 10 Ⅎ𝑑 𝑔 Fn ran 𝐺
156 nfv 1917 . . . . . . . . . . 11 Ⅎ𝑑(π‘”β€˜π‘™) ∈ 𝑙
157154, 156nfralw 3308 . . . . . . . . . 10 β„²π‘‘βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙
158 nfcv 2903 . . . . . . . . . . 11 Ⅎ𝑑𝑓
159 nfcv 2903 . . . . . . . . . . 11 Ⅎ𝑑(1...π‘š)
160158, 159, 154nff1o 6831 . . . . . . . . . 10 Ⅎ𝑑 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺
161155, 157, 160nf3an 1904 . . . . . . . . 9 Ⅎ𝑑(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)
162140, 161nfan 1902 . . . . . . . 8 Ⅎ𝑑(π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))
163139, 162nfan 1902 . . . . . . 7 Ⅎ𝑑(πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
164 nfv 1917 . . . . . . . . 9 Ⅎ𝑀 π‘š ∈ β„•
165 nfcv 2903 . . . . . . . . . . 11 Ⅎ𝑀𝑔
166165, 11nffn 6648 . . . . . . . . . 10 Ⅎ𝑀 𝑔 Fn ran 𝐺
167 nfv 1917 . . . . . . . . . . 11 Ⅎ𝑀(π‘”β€˜π‘™) ∈ 𝑙
16811, 167nfralw 3308 . . . . . . . . . 10 β„²π‘€βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙
169 nfcv 2903 . . . . . . . . . . 11 Ⅎ𝑀𝑓
170 nfcv 2903 . . . . . . . . . . 11 Ⅎ𝑀(1...π‘š)
171169, 170, 11nff1o 6831 . . . . . . . . . 10 Ⅎ𝑀 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺
172166, 168, 171nf3an 1904 . . . . . . . . 9 Ⅎ𝑀(𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)
173164, 172nfan 1902 . . . . . . . 8 Ⅎ𝑀(π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))
1748, 173nfan 1902 . . . . . . 7 Ⅎ𝑀(πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)))
1752, 128, 129, 130, 133, 136, 137, 138, 163, 174, 84stoweidlem27 44828 . . . . . 6 ((πœ‘ ∧ (π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺))) β†’ βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))))
176175ex 413 . . . . 5 (πœ‘ β†’ ((π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘)))))
1771762eximdv 1922 . . . 4 (πœ‘ β†’ (βˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘”βˆƒπ‘“βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘)))))
178177eximdv 1920 . . 3 (πœ‘ β†’ (βˆƒπ‘šβˆƒπ‘”βˆƒπ‘“(π‘š ∈ β„• ∧ (𝑔 Fn ran 𝐺 ∧ βˆ€π‘™ ∈ ran 𝐺(π‘”β€˜π‘™) ∈ 𝑙 ∧ 𝑓:(1...π‘š)–1-1-ontoβ†’ran 𝐺)) β†’ βˆƒπ‘šβˆƒπ‘”βˆƒπ‘“βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘)))))
179127, 178mpd 15 . 2 (πœ‘ β†’ βˆƒπ‘šβˆƒπ‘”βˆƒπ‘“βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))))
180 id 22 . . . 4 (βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))) β†’ βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))))
181180exlimivv 1935 . . 3 (βˆƒπ‘”βˆƒπ‘“βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))) β†’ βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))))
182181eximi 1837 . 2 (βˆƒπ‘šβˆƒπ‘”βˆƒπ‘“βˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))) β†’ βˆƒπ‘šβˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))))
183179, 182syl 17 1 (πœ‘ β†’ βˆƒπ‘šβˆƒπ‘ž(π‘š ∈ β„• ∧ (π‘ž:(1...π‘š)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...π‘š)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∨ wo 845   ∧ w3a 1087  βˆ€wal 1539   = wceq 1541  βˆƒwex 1781  β„²wnf 1785   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3945   βŠ† wss 3948  βˆ…c0 4322  βˆͺ cuni 4908   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411  Fincfn 8941  0cc0 11112  1c1 11113   < clt 11250   ≀ cle 11251  β„•cn 12214  ...cfz 13486  β™―chash 14292
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-hash 14293
This theorem is referenced by:  stoweidlem53  44854
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