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Theorem vdwlem11 16924
Description: Lemma for vdw 16927. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdw.r (πœ‘ β†’ 𝑅 ∈ Fin)
vdwlem9.k (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜2))
vdwlem9.s (πœ‘ β†’ βˆ€π‘  ∈ Fin βˆƒπ‘› ∈ β„• βˆ€π‘“ ∈ (𝑠 ↑m (1...𝑛))𝐾 MonoAP 𝑓)
Assertion
Ref Expression
vdwlem11 (πœ‘ β†’ βˆƒπ‘› ∈ β„• βˆ€π‘“ ∈ (𝑅 ↑m (1...𝑛))(𝐾 + 1) MonoAP 𝑓)
Distinct variable groups:   πœ‘,𝑛,𝑓   𝑓,𝑠,𝐾,𝑛   𝑅,𝑓,𝑛,𝑠   πœ‘,𝑓
Allowed substitution hint:   πœ‘(𝑠)

Proof of Theorem vdwlem11
Dummy variables π‘Ž 𝑑 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdw.r . . 3 (πœ‘ β†’ 𝑅 ∈ Fin)
2 vdwlem9.k . . 3 (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜2))
3 vdwlem9.s . . 3 (πœ‘ β†’ βˆ€π‘  ∈ Fin βˆƒπ‘› ∈ β„• βˆ€π‘“ ∈ (𝑠 ↑m (1...𝑛))𝐾 MonoAP 𝑓)
4 hashcl 14316 . . . . 5 (𝑅 ∈ Fin β†’ (β™―β€˜π‘…) ∈ β„•0)
51, 4syl 17 . . . 4 (πœ‘ β†’ (β™―β€˜π‘…) ∈ β„•0)
6 nn0p1nn 12511 . . . 4 ((β™―β€˜π‘…) ∈ β„•0 β†’ ((β™―β€˜π‘…) + 1) ∈ β„•)
75, 6syl 17 . . 3 (πœ‘ β†’ ((β™―β€˜π‘…) + 1) ∈ β„•)
81, 2, 3, 7vdwlem10 16923 . 2 (πœ‘ β†’ βˆƒπ‘› ∈ β„• βˆ€π‘“ ∈ (𝑅 ↑m (1...𝑛))(⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
91adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑅 ∈ Fin)
10 ovex 7442 . . . . . . 7 (1...𝑛) ∈ V
11 elmapg 8833 . . . . . . 7 ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) β†’ (𝑓 ∈ (𝑅 ↑m (1...𝑛)) ↔ 𝑓:(1...𝑛)βŸΆπ‘…))
129, 10, 11sylancl 587 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (𝑓 ∈ (𝑅 ↑m (1...𝑛)) ↔ 𝑓:(1...𝑛)βŸΆπ‘…))
1312biimpa 478 . . . . 5 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) β†’ 𝑓:(1...𝑛)βŸΆπ‘…)
145nn0red 12533 . . . . . . . . . . 11 (πœ‘ β†’ (β™―β€˜π‘…) ∈ ℝ)
1514ltp1d 12144 . . . . . . . . . 10 (πœ‘ β†’ (β™―β€˜π‘…) < ((β™―β€˜π‘…) + 1))
16 peano2re 11387 . . . . . . . . . . . 12 ((β™―β€˜π‘…) ∈ ℝ β†’ ((β™―β€˜π‘…) + 1) ∈ ℝ)
1714, 16syl 17 . . . . . . . . . . 11 (πœ‘ β†’ ((β™―β€˜π‘…) + 1) ∈ ℝ)
1814, 17ltnled 11361 . . . . . . . . . 10 (πœ‘ β†’ ((β™―β€˜π‘…) < ((β™―β€˜π‘…) + 1) ↔ Β¬ ((β™―β€˜π‘…) + 1) ≀ (β™―β€˜π‘…)))
1915, 18mpbid 231 . . . . . . . . 9 (πœ‘ β†’ Β¬ ((β™―β€˜π‘…) + 1) ≀ (β™―β€˜π‘…))
2019adantr 482 . . . . . . . 8 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ Β¬ ((β™―β€˜π‘…) + 1) ≀ (β™―β€˜π‘…))
21 eluz2nn 12868 . . . . . . . . . . . . 13 (𝐾 ∈ (β„€β‰₯β€˜2) β†’ 𝐾 ∈ β„•)
222, 21syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐾 ∈ β„•)
2322adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ 𝐾 ∈ β„•)
2423nnnn0d 12532 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ 𝐾 ∈ β„•0)
25 simprr 772 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ 𝑓:(1...𝑛)βŸΆπ‘…)
267adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ ((β™―β€˜π‘…) + 1) ∈ β„•)
27 eqid 2733 . . . . . . . . . 10 (1...((β™―β€˜π‘…) + 1)) = (1...((β™―β€˜π‘…) + 1))
2810, 24, 25, 26, 27vdwpc 16913 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ (⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 ↔ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1)))(βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = ((β™―β€˜π‘…) + 1))))
291ad3antrrr 729 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ 𝑅 ∈ Fin)
3025ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) β†’ 𝑓:(1...𝑛)βŸΆπ‘…)
3125ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) ∧ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ 𝑓:(1...𝑛)βŸΆπ‘…)
32 simpr 486 . . . . . . . . . . . . . . . . . . . . . . 23 (((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) ∧ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}))
33 cnvimass 6081 . . . . . . . . . . . . . . . . . . . . . . 23 (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) βŠ† dom 𝑓
3432, 33sstrdi 3995 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) ∧ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† dom 𝑓)
3531, 34fssdmd 6737 . . . . . . . . . . . . . . . . . . . . 21 (((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) ∧ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (1...𝑛))
3622ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) β†’ 𝐾 ∈ β„•)
37 simplrl 776 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) β†’ π‘Ž ∈ β„•)
38 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) β†’ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))
39 nnex 12218 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 β„• ∈ V
40 ovex 7442 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1...((β™―β€˜π‘…) + 1)) ∈ V
4139, 40elmap 8865 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))) ↔ 𝑑:(1...((β™―β€˜π‘…) + 1))βŸΆβ„•)
4238, 41sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) β†’ 𝑑:(1...((β™―β€˜π‘…) + 1))βŸΆβ„•)
4342ffvelcdmda 7087 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) β†’ (π‘‘β€˜π‘–) ∈ β„•)
4437, 43nnaddcld 12264 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) β†’ (π‘Ž + (π‘‘β€˜π‘–)) ∈ β„•)
45 vdwapid1 16908 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐾 ∈ β„• ∧ (π‘Ž + (π‘‘β€˜π‘–)) ∈ β„• ∧ (π‘‘β€˜π‘–) ∈ β„•) β†’ (π‘Ž + (π‘‘β€˜π‘–)) ∈ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)))
4636, 44, 43, 45syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . 22 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) β†’ (π‘Ž + (π‘‘β€˜π‘–)) ∈ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)))
4746adantr 482 . . . . . . . . . . . . . . . . . . . . 21 (((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) ∧ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ (π‘Ž + (π‘‘β€˜π‘–)) ∈ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)))
4835, 47sseldd 3984 . . . . . . . . . . . . . . . . . . . 20 (((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) ∧ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ (π‘Ž + (π‘‘β€˜π‘–)) ∈ (1...𝑛))
4948ex 414 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) β†’ (((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) β†’ (π‘Ž + (π‘‘β€˜π‘–)) ∈ (1...𝑛)))
50 ffvelcdm 7084 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑛)βŸΆπ‘… ∧ (π‘Ž + (π‘‘β€˜π‘–)) ∈ (1...𝑛)) β†’ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))) ∈ 𝑅)
5130, 49, 50syl6an 683 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ 𝑖 ∈ (1...((β™―β€˜π‘…) + 1))) β†’ (((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) β†’ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))) ∈ 𝑅))
5251ralimdva 3168 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) β†’ (βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) β†’ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))) ∈ 𝑅))
5352imp 408 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))) ∈ 𝑅)
54 eqid 2733 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) = (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))
5554fmpt 7110 . . . . . . . . . . . . . . . 16 (βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))) ∈ 𝑅 ↔ (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))):(1...((β™―β€˜π‘…) + 1))βŸΆπ‘…)
5653, 55sylib 217 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))):(1...((β™―β€˜π‘…) + 1))βŸΆπ‘…)
5756frnd 6726 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) βŠ† 𝑅)
58 ssdomg 8996 . . . . . . . . . . . . . 14 (𝑅 ∈ Fin β†’ (ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) βŠ† 𝑅 β†’ ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) β‰Ό 𝑅))
5929, 57, 58sylc 65 . . . . . . . . . . . . 13 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) β‰Ό 𝑅)
6029, 57ssfid 9267 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) ∈ Fin)
61 hashdom 14339 . . . . . . . . . . . . . 14 ((ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) ∈ Fin ∧ 𝑅 ∈ Fin) β†’ ((β™―β€˜ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) ≀ (β™―β€˜π‘…) ↔ ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) β‰Ό 𝑅))
6260, 29, 61syl2anc 585 . . . . . . . . . . . . 13 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ ((β™―β€˜ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) ≀ (β™―β€˜π‘…) ↔ ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))) β‰Ό 𝑅))
6359, 62mpbird 257 . . . . . . . . . . . 12 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ (β™―β€˜ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) ≀ (β™―β€˜π‘…))
64 breq1 5152 . . . . . . . . . . . 12 ((β™―β€˜ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = ((β™―β€˜π‘…) + 1) β†’ ((β™―β€˜ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) ≀ (β™―β€˜π‘…) ↔ ((β™―β€˜π‘…) + 1) ≀ (β™―β€˜π‘…)))
6563, 64syl5ibcom 244 . . . . . . . . . . 11 ((((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) ∧ βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))})) β†’ ((β™―β€˜ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = ((β™―β€˜π‘…) + 1) β†’ ((β™―β€˜π‘…) + 1) ≀ (β™―β€˜π‘…)))
6665expimpd 455 . . . . . . . . . 10 (((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1))))) β†’ ((βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = ((β™―β€˜π‘…) + 1)) β†’ ((β™―β€˜π‘…) + 1) ≀ (β™―β€˜π‘…)))
6766rexlimdvva 3212 . . . . . . . . 9 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ (βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m (1...((β™―β€˜π‘…) + 1)))(βˆ€π‘– ∈ (1...((β™―β€˜π‘…) + 1))((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...((β™―β€˜π‘…) + 1)) ↦ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))))) = ((β™―β€˜π‘…) + 1)) β†’ ((β™―β€˜π‘…) + 1) ≀ (β™―β€˜π‘…)))
6828, 67sylbid 239 . . . . . . . 8 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ (⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 β†’ ((β™―β€˜π‘…) + 1) ≀ (β™―β€˜π‘…)))
6920, 68mtod 197 . . . . . . 7 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ Β¬ ⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓)
70 biorf 936 . . . . . . 7 (Β¬ ⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 β†’ ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7169, 70syl 17 . . . . . 6 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7271anassrs 469 . . . . 5 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ 𝑓:(1...𝑛)βŸΆπ‘…) β†’ ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7313, 72syldan 592 . . . 4 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) β†’ ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7473ralbidva 3176 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (βˆ€π‘“ ∈ (𝑅 ↑m (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ βˆ€π‘“ ∈ (𝑅 ↑m (1...𝑛))(⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7574rexbidva 3177 . 2 (πœ‘ β†’ (βˆƒπ‘› ∈ β„• βˆ€π‘“ ∈ (𝑅 ↑m (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ βˆƒπ‘› ∈ β„• βˆ€π‘“ ∈ (𝑅 ↑m (1...𝑛))(⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
768, 75mpbird 257 1 (πœ‘ β†’ βˆƒπ‘› ∈ β„• βˆ€π‘“ ∈ (𝑅 ↑m (1...𝑛))(𝐾 + 1) MonoAP 𝑓)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  {csn 4629  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820   β‰Ό cdom 8937  Fincfn 8939  β„cr 11109  1c1 11111   + caddc 11113   < clt 11248   ≀ cle 11249  β„•cn 12212  2c2 12267  β„•0cn0 12472  β„€β‰₯cuz 12822  ...cfz 13484  β™―chash 14290  APcvdwa 16898   MonoAP cvdwm 16899   PolyAP cvdwp 16900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-hash 14291  df-vdwap 16901  df-vdwmc 16902  df-vdwpc 16903
This theorem is referenced by:  vdwlem13  16926
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