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Theorem vdwlem11 16926
Description: Lemma for vdw 16929. (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 14318 . . . . 5 (𝑅 ∈ Fin β†’ (β™―β€˜π‘…) ∈ β„•0)
51, 4syl 17 . . . 4 (πœ‘ β†’ (β™―β€˜π‘…) ∈ β„•0)
6 nn0p1nn 12513 . . . 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 16925 . 2 (πœ‘ β†’ βˆƒπ‘› ∈ β„• βˆ€π‘“ ∈ (𝑅 ↑m (1...𝑛))(⟨((β™―β€˜π‘…) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
91adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑅 ∈ Fin)
10 ovex 7444 . . . . . . 7 (1...𝑛) ∈ V
11 elmapg 8835 . . . . . . 7 ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) β†’ (𝑓 ∈ (𝑅 ↑m (1...𝑛)) ↔ 𝑓:(1...𝑛)βŸΆπ‘…))
129, 10, 11sylancl 586 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (𝑓 ∈ (𝑅 ↑m (1...𝑛)) ↔ 𝑓:(1...𝑛)βŸΆπ‘…))
1312biimpa 477 . . . . 5 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) β†’ 𝑓:(1...𝑛)βŸΆπ‘…)
145nn0red 12535 . . . . . . . . . . 11 (πœ‘ β†’ (β™―β€˜π‘…) ∈ ℝ)
1514ltp1d 12146 . . . . . . . . . 10 (πœ‘ β†’ (β™―β€˜π‘…) < ((β™―β€˜π‘…) + 1))
16 peano2re 11389 . . . . . . . . . . . 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 11363 . . . . . . . . . 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 481 . . . . . . . 8 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ Β¬ ((β™―β€˜π‘…) + 1) ≀ (β™―β€˜π‘…))
21 eluz2nn 12870 . . . . . . . . . . . . 13 (𝐾 ∈ (β„€β‰₯β€˜2) β†’ 𝐾 ∈ β„•)
222, 21syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐾 ∈ β„•)
2322adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ 𝐾 ∈ β„•)
2423nnnn0d 12534 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ 𝐾 ∈ β„•0)
25 simprr 771 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ 𝑓:(1...𝑛)βŸΆπ‘…)
267adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ 𝑓:(1...𝑛)βŸΆπ‘…)) β†’ ((β™―β€˜π‘…) + 1) ∈ β„•)
27 eqid 2732 . . . . . . . . . 10 (1...((β™―β€˜π‘…) + 1)) = (1...((β™―β€˜π‘…) + 1))
2810, 24, 25, 26, 27vdwpc 16915 . . . . . . . . 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 728 . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . 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 6080 . . . . . . . . . . . . . . . . . . . . . . 23 (◑𝑓 β€œ {(π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–)))}) βŠ† dom 𝑓
3432, 33sstrdi 3994 . . . . . . . . . . . . . . . . . . . . . 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 6736 . . . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . . . . . . . . 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 12220 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 β„• ∈ V
40 ovex 7444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1...((β™―β€˜π‘…) + 1)) ∈ V
4139, 40elmap 8867 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . . . . . . . . . . . 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 12266 . . . . . . . . . . . . . . . . . . . . . . 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 16910 . . . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . 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 7083 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑛)βŸΆπ‘… ∧ (π‘Ž + (π‘‘β€˜π‘–)) ∈ (1...𝑛)) β†’ (π‘“β€˜(π‘Ž + (π‘‘β€˜π‘–))) ∈ 𝑅)
5130, 49, 50syl6an 682 . . . . . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . 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 7111 . . . . . . . . . . . . . . . 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 6725 . . . . . . . . . . . . . 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 8998 . . . . . . . . . . . . . 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 9269 . . . . . . . . . . . . . 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 14341 . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 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 256 . . . . . . . . . . . 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 5151 . . . . . . . . . . . 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 454 . . . . . . . . . 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 3211 . . . . . . . . 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 935 . . . . . . 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 468 . . . . 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 591 . . . 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 3175 . . 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 3176 . 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 256 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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3948  {csn 4628  βŸ¨cop 4634   class class class wbr 5148   ↦ cmpt 5231  β—‘ccnv 5675  dom cdm 5676  ran crn 5677   β€œ cima 5679  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ↑m cmap 8822   β‰Ό cdom 8939  Fincfn 8941  β„cr 11111  1c1 11113   + caddc 11115   < clt 11250   ≀ cle 11251  β„•cn 12214  2c2 12269  β„•0cn0 12474  β„€β‰₯cuz 12824  ...cfz 13486  β™―chash 14292  APcvdwa 16900   MonoAP cvdwm 16901   PolyAP cvdwp 16902
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-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  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-2 12277  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-rp 12977  df-fz 13487  df-hash 14293  df-vdwap 16903  df-vdwmc 16904  df-vdwpc 16905
This theorem is referenced by:  vdwlem13  16928
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