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Theorem vdwlem7 16920
Description: Lemma for vdw 16927. (Contributed by Mario Carneiro, 12-Sep-2014.)
Hypotheses
Ref Expression
vdwlem3.v (πœ‘ β†’ 𝑉 ∈ β„•)
vdwlem3.w (πœ‘ β†’ π‘Š ∈ β„•)
vdwlem4.r (πœ‘ β†’ 𝑅 ∈ Fin)
vdwlem4.h (πœ‘ β†’ 𝐻:(1...(π‘Š Β· (2 Β· 𝑉)))βŸΆπ‘…)
vdwlem4.f 𝐹 = (π‘₯ ∈ (1...𝑉) ↦ (𝑦 ∈ (1...π‘Š) ↦ (π»β€˜(𝑦 + (π‘Š Β· ((π‘₯ βˆ’ 1) + 𝑉))))))
vdwlem7.m (πœ‘ β†’ 𝑀 ∈ β„•)
vdwlem7.g (πœ‘ β†’ 𝐺:(1...π‘Š)βŸΆπ‘…)
vdwlem7.k (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜2))
vdwlem7.a (πœ‘ β†’ 𝐴 ∈ β„•)
vdwlem7.d (πœ‘ β†’ 𝐷 ∈ β„•)
vdwlem7.s (πœ‘ β†’ (𝐴(APβ€˜πΎ)𝐷) βŠ† (◑𝐹 β€œ {𝐺}))
Assertion
Ref Expression
vdwlem7 (πœ‘ β†’ (βŸ¨π‘€, 𝐾⟩ PolyAP 𝐺 β†’ (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝐺,𝑦   π‘₯,𝐾,𝑦   πœ‘,π‘₯,𝑦   π‘₯,𝑅,𝑦   π‘₯,𝐻,𝑦   π‘₯,𝑀,𝑦   π‘₯,𝐷,𝑦   π‘₯,π‘Š,𝑦   π‘₯,𝑉,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)

Proof of Theorem vdwlem7
Dummy variables π‘˜ π‘Ž 𝑑 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7442 . . 3 (1...π‘Š) ∈ V
2 2nn0 12489 . . . 4 2 ∈ β„•0
3 vdwlem7.k . . . 4 (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜2))
4 eluznn0 12901 . . . 4 ((2 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜2)) β†’ 𝐾 ∈ β„•0)
52, 3, 4sylancr 588 . . 3 (πœ‘ β†’ 𝐾 ∈ β„•0)
6 vdwlem7.g . . 3 (πœ‘ β†’ 𝐺:(1...π‘Š)βŸΆπ‘…)
7 vdwlem7.m . . 3 (πœ‘ β†’ 𝑀 ∈ β„•)
8 eqid 2733 . . 3 (1...𝑀) = (1...𝑀)
91, 5, 6, 7, 8vdwpc 16913 . 2 (πœ‘ β†’ (βŸ¨π‘€, 𝐾⟩ PolyAP 𝐺 ↔ βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m (1...𝑀))(βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)))
10 vdwlem3.v . . . . . 6 (πœ‘ β†’ 𝑉 ∈ β„•)
1110ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝑉 ∈ β„•)
12 vdwlem3.w . . . . . 6 (πœ‘ β†’ π‘Š ∈ β„•)
1312ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ π‘Š ∈ β„•)
14 vdwlem4.r . . . . . 6 (πœ‘ β†’ 𝑅 ∈ Fin)
1514ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝑅 ∈ Fin)
16 vdwlem4.h . . . . . 6 (πœ‘ β†’ 𝐻:(1...(π‘Š Β· (2 Β· 𝑉)))βŸΆπ‘…)
1716ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝐻:(1...(π‘Š Β· (2 Β· 𝑉)))βŸΆπ‘…)
18 vdwlem4.f . . . . 5 𝐹 = (π‘₯ ∈ (1...𝑉) ↦ (𝑦 ∈ (1...π‘Š) ↦ (π»β€˜(𝑦 + (π‘Š Β· ((π‘₯ βˆ’ 1) + 𝑉))))))
197ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝑀 ∈ β„•)
206ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝐺:(1...π‘Š)βŸΆπ‘…)
213ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝐾 ∈ (β„€β‰₯β€˜2))
22 vdwlem7.a . . . . . 6 (πœ‘ β†’ 𝐴 ∈ β„•)
2322ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝐴 ∈ β„•)
24 vdwlem7.d . . . . . 6 (πœ‘ β†’ 𝐷 ∈ β„•)
2524ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝐷 ∈ β„•)
26 vdwlem7.s . . . . . 6 (πœ‘ β†’ (𝐴(APβ€˜πΎ)𝐷) βŠ† (◑𝐹 β€œ {𝐺}))
2726ad2antrr 725 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ (𝐴(APβ€˜πΎ)𝐷) βŠ† (◑𝐹 β€œ {𝐺}))
28 simplrl 776 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ π‘Ž ∈ β„•)
29 simplrr 777 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝑑 ∈ (β„• ↑m (1...𝑀)))
30 nnex 12218 . . . . . . 7 β„• ∈ V
31 ovex 7442 . . . . . . 7 (1...𝑀) ∈ V
3230, 31elmap 8865 . . . . . 6 (𝑑 ∈ (β„• ↑m (1...𝑀)) ↔ 𝑑:(1...𝑀)βŸΆβ„•)
3329, 32sylib 217 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ 𝑑:(1...𝑀)βŸΆβ„•)
34 simprl 770 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}))
35 fveq2 6892 . . . . . . . . . 10 (𝑖 = π‘˜ β†’ (π‘‘β€˜π‘–) = (π‘‘β€˜π‘˜))
3635oveq2d 7425 . . . . . . . . 9 (𝑖 = π‘˜ β†’ (π‘Ž + (π‘‘β€˜π‘–)) = (π‘Ž + (π‘‘β€˜π‘˜)))
3736, 35oveq12d 7427 . . . . . . . 8 (𝑖 = π‘˜ β†’ ((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) = ((π‘Ž + (π‘‘β€˜π‘˜))(APβ€˜πΎ)(π‘‘β€˜π‘˜)))
3836fveq2d 6896 . . . . . . . . . 10 (𝑖 = π‘˜ β†’ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))) = (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜))))
3938sneqd 4641 . . . . . . . . 9 (𝑖 = π‘˜ β†’ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))} = {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜)))})
4039imaeq2d 6060 . . . . . . . 8 (𝑖 = π‘˜ β†’ (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) = (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜)))}))
4137, 40sseq12d 4016 . . . . . . 7 (𝑖 = π‘˜ β†’ (((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ↔ ((π‘Ž + (π‘‘β€˜π‘˜))(APβ€˜πΎ)(π‘‘β€˜π‘˜)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜)))})))
4241cbvralvw 3235 . . . . . 6 (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ↔ βˆ€π‘˜ ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘˜))(APβ€˜πΎ)(π‘‘β€˜π‘˜)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜)))}))
4334, 42sylib 217 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ βˆ€π‘˜ ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘˜))(APβ€˜πΎ)(π‘‘β€˜π‘˜)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜)))}))
4438cbvmptv 5262 . . . . 5 (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))) = (π‘˜ ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜))))
45 simprr 772 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)
46 eqid 2733 . . . . 5 (π‘Ž + (π‘Š Β· ((𝐴 + (𝑉 βˆ’ 𝐷)) βˆ’ 1))) = (π‘Ž + (π‘Š Β· ((𝐴 + (𝑉 βˆ’ 𝐷)) βˆ’ 1)))
47 eqid 2733 . . . . 5 (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (π‘‘β€˜π‘—)) + (π‘Š Β· 𝐷))) = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (π‘‘β€˜π‘—)) + (π‘Š Β· 𝐷)))
4811, 13, 15, 17, 18, 19, 20, 21, 23, 25, 27, 28, 33, 43, 44, 45, 46, 47vdwlem6 16919 . . . 4 (((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) ∧ (βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀)) β†’ (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
4948ex 414 . . 3 ((πœ‘ ∧ (π‘Ž ∈ β„• ∧ 𝑑 ∈ (β„• ↑m (1...𝑀)))) β†’ ((βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀) β†’ (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
5049rexlimdvva 3212 . 2 (πœ‘ β†’ (βˆƒπ‘Ž ∈ β„• βˆƒπ‘‘ ∈ (β„• ↑m (1...𝑀))(βˆ€π‘– ∈ (1...𝑀)((π‘Ž + (π‘‘β€˜π‘–))(APβ€˜πΎ)(π‘‘β€˜π‘–)) βŠ† (◑𝐺 β€œ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))}) ∧ (β™―β€˜ran (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))))) = 𝑀) β†’ (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
519, 50sylbid 239 1 (πœ‘ β†’ (βŸ¨π‘€, 𝐾⟩ PolyAP 𝐺 β†’ (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   βŠ† wss 3949  ifcif 4529  {csn 4629  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232  β—‘ccnv 5676  ran crn 5678   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  Fincfn 8939  0cc0 11110  1c1 11111   + caddc 11113   Β· cmul 11115   βˆ’ cmin 11444  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-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-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:  vdwlem9  16922
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