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Theorem vdwlem7 16924
Description: Lemma for vdw 16931. (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 7444 . . 3 (1...π‘Š) ∈ V
2 2nn0 12493 . . . 4 2 ∈ β„•0
3 vdwlem7.k . . . 4 (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜2))
4 eluznn0 12905 . . . 4 ((2 ∈ β„•0 ∧ 𝐾 ∈ (β„€β‰₯β€˜2)) β†’ 𝐾 ∈ β„•0)
52, 3, 4sylancr 585 . . 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 2730 . . 3 (1...𝑀) = (1...𝑀)
91, 5, 6, 7, 8vdwpc 16917 . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 773 . . . . 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 774 . . . . . 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 12222 . . . . . . 7 β„• ∈ V
31 ovex 7444 . . . . . . 7 (1...𝑀) ∈ V
3230, 31elmap 8867 . . . . . 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 767 . . . . . 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 6890 . . . . . . . . . 10 (𝑖 = π‘˜ β†’ (π‘‘β€˜π‘–) = (π‘‘β€˜π‘˜))
3635oveq2d 7427 . . . . . . . . 9 (𝑖 = π‘˜ β†’ (π‘Ž + (π‘‘β€˜π‘–)) = (π‘Ž + (π‘‘β€˜π‘˜)))
3736, 35oveq12d 7429 . . . . . . . 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 6894 . . . . . . . . . 10 (𝑖 = π‘˜ β†’ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–))) = (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜))))
3938sneqd 4639 . . . . . . . . 9 (𝑖 = π‘˜ β†’ {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))} = {(πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜)))})
4039imaeq2d 6058 . . . . . . . 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 4014 . . . . . . 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 3232 . . . . . 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 5260 . . . . 5 (𝑖 ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘–)))) = (π‘˜ ∈ (1...𝑀) ↦ (πΊβ€˜(π‘Ž + (π‘‘β€˜π‘˜))))
45 simprr 769 . . . . 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 2730 . . . . 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 2730 . . . . 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 16923 . . . 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 411 . . 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 3209 . 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 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   βŠ† wss 3947  ifcif 4527  {csn 4627  βŸ¨cop 4633   class class class wbr 5147   ↦ cmpt 5230  β—‘ccnv 5674  ran crn 5676   β€œ cima 5678  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  Fincfn 8941  0cc0 11112  1c1 11113   + caddc 11115   Β· cmul 11117   βˆ’ cmin 11448  β„•cn 12216  2c2 12271  β„•0cn0 12476  β„€β‰₯cuz 12826  ...cfz 13488  β™―chash 14294  APcvdwa 16902   MonoAP cvdwm 16903   PolyAP cvdwp 16904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  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 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  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-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-fz 13489  df-hash 14295  df-vdwap 16905  df-vdwmc 16906  df-vdwpc 16907
This theorem is referenced by:  vdwlem9  16926
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