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Theorem plydivlem4 26247
Description: Lemma for plydivex 26248. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
plydiv.pl ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)
plydiv.tm ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ Β· 𝑦) ∈ 𝑆)
plydiv.rc ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ π‘₯ β‰  0)) β†’ (1 / π‘₯) ∈ 𝑆)
plydiv.m1 (πœ‘ β†’ -1 ∈ 𝑆)
plydiv.f (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜π‘†))
plydiv.g (πœ‘ β†’ 𝐺 ∈ (Polyβ€˜π‘†))
plydiv.z (πœ‘ β†’ 𝐺 β‰  0𝑝)
plydiv.r 𝑅 = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· π‘ž))
plydiv.d (πœ‘ β†’ 𝐷 ∈ β„•0)
plydiv.e (πœ‘ β†’ (𝑀 βˆ’ 𝑁) = 𝐷)
plydiv.fz (πœ‘ β†’ 𝐹 β‰  0𝑝)
plydiv.u π‘ˆ = (𝑓 ∘f βˆ’ (𝐺 ∘f Β· 𝑝))
plydiv.h 𝐻 = (𝑧 ∈ β„‚ ↦ (((π΄β€˜π‘€) / (π΅β€˜π‘)) Β· (𝑧↑𝐷)))
plydiv.al (πœ‘ β†’ βˆ€π‘“ ∈ (Polyβ€˜π‘†)((𝑓 = 0𝑝 ∨ ((degβ€˜π‘“) βˆ’ 𝑁) < 𝐷) β†’ βˆƒπ‘ ∈ (Polyβ€˜π‘†)(π‘ˆ = 0𝑝 ∨ (degβ€˜π‘ˆ) < 𝑁)))
plydiv.a 𝐴 = (coeffβ€˜πΉ)
plydiv.b 𝐡 = (coeffβ€˜πΊ)
plydiv.m 𝑀 = (degβ€˜πΉ)
plydiv.n 𝑁 = (degβ€˜πΊ)
Assertion
Ref Expression
plydivlem4 (πœ‘ β†’ βˆƒπ‘ž ∈ (Polyβ€˜π‘†)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < 𝑁))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐴   𝑓,𝑝,π‘ž,π‘₯,𝑦,𝑧,𝐹   𝑓,𝐻,𝑝,π‘ž,π‘₯,𝑦,𝑧   πœ‘,π‘₯,𝑦,𝑧   π‘₯,𝐡,𝑦,𝑧   𝐷,𝑓,𝑧   π‘₯,𝑀,𝑦,𝑧   𝑓,𝑁,𝑝,π‘ž,π‘₯,𝑦,𝑧   𝑓,𝐺,𝑝,π‘ž,π‘₯,𝑦,𝑧   𝑅,𝑓,𝑝,π‘₯,𝑦   𝑆,𝑓,𝑝,π‘ž,π‘₯,𝑦,𝑧   πœ‘,𝑝
Allowed substitution hints:   πœ‘(𝑓,π‘ž)   𝐴(𝑓,π‘ž,𝑝)   𝐡(𝑓,π‘ž,𝑝)   𝐷(π‘₯,𝑦,π‘ž,𝑝)   𝑅(𝑧,π‘ž)   π‘ˆ(π‘₯,𝑦,𝑧,𝑓,π‘ž,𝑝)   𝑀(𝑓,π‘ž,𝑝)

Proof of Theorem plydivlem4
StepHypRef Expression
1 plydiv.f . . . . . . 7 (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜π‘†))
2 plybss 26144 . . . . . . 7 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝑆 βŠ† β„‚)
31, 2syl 17 . . . . . 6 (πœ‘ β†’ 𝑆 βŠ† β„‚)
4 plydiv.pl . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)
5 plydiv.tm . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ Β· 𝑦) ∈ 𝑆)
6 plydiv.rc . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ π‘₯ β‰  0)) β†’ (1 / π‘₯) ∈ 𝑆)
7 plydiv.m1 . . . . . . . . . . . 12 (πœ‘ β†’ -1 ∈ 𝑆)
84, 5, 6, 7plydivlem1 26244 . . . . . . . . . . 11 (πœ‘ β†’ 0 ∈ 𝑆)
9 plydiv.a . . . . . . . . . . . 12 𝐴 = (coeffβ€˜πΉ)
109coef2 26181 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 0 ∈ 𝑆) β†’ 𝐴:β„•0βŸΆπ‘†)
111, 8, 10syl2anc 582 . . . . . . . . . 10 (πœ‘ β†’ 𝐴:β„•0βŸΆπ‘†)
12 plydiv.m . . . . . . . . . . 11 𝑀 = (degβ€˜πΉ)
13 dgrcl 26183 . . . . . . . . . . . 12 (𝐹 ∈ (Polyβ€˜π‘†) β†’ (degβ€˜πΉ) ∈ β„•0)
141, 13syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (degβ€˜πΉ) ∈ β„•0)
1512, 14eqeltrid 2829 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 ∈ β„•0)
1611, 15ffvelcdmd 7089 . . . . . . . . 9 (πœ‘ β†’ (π΄β€˜π‘€) ∈ 𝑆)
173, 16sseldd 3973 . . . . . . . 8 (πœ‘ β†’ (π΄β€˜π‘€) ∈ β„‚)
18 plydiv.g . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 ∈ (Polyβ€˜π‘†))
19 plydiv.b . . . . . . . . . . . 12 𝐡 = (coeffβ€˜πΊ)
2019coef2 26181 . . . . . . . . . . 11 ((𝐺 ∈ (Polyβ€˜π‘†) ∧ 0 ∈ 𝑆) β†’ 𝐡:β„•0βŸΆπ‘†)
2118, 8, 20syl2anc 582 . . . . . . . . . 10 (πœ‘ β†’ 𝐡:β„•0βŸΆπ‘†)
22 plydiv.n . . . . . . . . . . 11 𝑁 = (degβ€˜πΊ)
23 dgrcl 26183 . . . . . . . . . . . 12 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (degβ€˜πΊ) ∈ β„•0)
2418, 23syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (degβ€˜πΊ) ∈ β„•0)
2522, 24eqeltrid 2829 . . . . . . . . . 10 (πœ‘ β†’ 𝑁 ∈ β„•0)
2621, 25ffvelcdmd 7089 . . . . . . . . 9 (πœ‘ β†’ (π΅β€˜π‘) ∈ 𝑆)
273, 26sseldd 3973 . . . . . . . 8 (πœ‘ β†’ (π΅β€˜π‘) ∈ β„‚)
28 plydiv.z . . . . . . . . 9 (πœ‘ β†’ 𝐺 β‰  0𝑝)
2922, 19dgreq0 26216 . . . . . . . . . . 11 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (𝐺 = 0𝑝 ↔ (π΅β€˜π‘) = 0))
3018, 29syl 17 . . . . . . . . . 10 (πœ‘ β†’ (𝐺 = 0𝑝 ↔ (π΅β€˜π‘) = 0))
3130necon3bid 2975 . . . . . . . . 9 (πœ‘ β†’ (𝐺 β‰  0𝑝 ↔ (π΅β€˜π‘) β‰  0))
3228, 31mpbid 231 . . . . . . . 8 (πœ‘ β†’ (π΅β€˜π‘) β‰  0)
3317, 27, 32divrecd 12021 . . . . . . 7 (πœ‘ β†’ ((π΄β€˜π‘€) / (π΅β€˜π‘)) = ((π΄β€˜π‘€) Β· (1 / (π΅β€˜π‘))))
34 fvex 6904 . . . . . . . . . . 11 (π΅β€˜π‘) ∈ V
35 eleq1 2813 . . . . . . . . . . . . . 14 (π‘₯ = (π΅β€˜π‘) β†’ (π‘₯ ∈ 𝑆 ↔ (π΅β€˜π‘) ∈ 𝑆))
36 neeq1 2993 . . . . . . . . . . . . . 14 (π‘₯ = (π΅β€˜π‘) β†’ (π‘₯ β‰  0 ↔ (π΅β€˜π‘) β‰  0))
3735, 36anbi12d 630 . . . . . . . . . . . . 13 (π‘₯ = (π΅β€˜π‘) β†’ ((π‘₯ ∈ 𝑆 ∧ π‘₯ β‰  0) ↔ ((π΅β€˜π‘) ∈ 𝑆 ∧ (π΅β€˜π‘) β‰  0)))
3837anbi2d 628 . . . . . . . . . . . 12 (π‘₯ = (π΅β€˜π‘) β†’ ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ π‘₯ β‰  0)) ↔ (πœ‘ ∧ ((π΅β€˜π‘) ∈ 𝑆 ∧ (π΅β€˜π‘) β‰  0))))
39 oveq2 7423 . . . . . . . . . . . . 13 (π‘₯ = (π΅β€˜π‘) β†’ (1 / π‘₯) = (1 / (π΅β€˜π‘)))
4039eleq1d 2810 . . . . . . . . . . . 12 (π‘₯ = (π΅β€˜π‘) β†’ ((1 / π‘₯) ∈ 𝑆 ↔ (1 / (π΅β€˜π‘)) ∈ 𝑆))
4138, 40imbi12d 343 . . . . . . . . . . 11 (π‘₯ = (π΅β€˜π‘) β†’ (((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ π‘₯ β‰  0)) β†’ (1 / π‘₯) ∈ 𝑆) ↔ ((πœ‘ ∧ ((π΅β€˜π‘) ∈ 𝑆 ∧ (π΅β€˜π‘) β‰  0)) β†’ (1 / (π΅β€˜π‘)) ∈ 𝑆)))
4234, 41, 6vtocl 3536 . . . . . . . . . 10 ((πœ‘ ∧ ((π΅β€˜π‘) ∈ 𝑆 ∧ (π΅β€˜π‘) β‰  0)) β†’ (1 / (π΅β€˜π‘)) ∈ 𝑆)
4342ex 411 . . . . . . . . 9 (πœ‘ β†’ (((π΅β€˜π‘) ∈ 𝑆 ∧ (π΅β€˜π‘) β‰  0) β†’ (1 / (π΅β€˜π‘)) ∈ 𝑆))
4426, 32, 43mp2and 697 . . . . . . . 8 (πœ‘ β†’ (1 / (π΅β€˜π‘)) ∈ 𝑆)
455, 16, 44caovcld 7610 . . . . . . 7 (πœ‘ β†’ ((π΄β€˜π‘€) Β· (1 / (π΅β€˜π‘))) ∈ 𝑆)
4633, 45eqeltrd 2825 . . . . . 6 (πœ‘ β†’ ((π΄β€˜π‘€) / (π΅β€˜π‘)) ∈ 𝑆)
47 plydiv.d . . . . . 6 (πœ‘ β†’ 𝐷 ∈ β„•0)
48 plydiv.h . . . . . . 7 𝐻 = (𝑧 ∈ β„‚ ↦ (((π΄β€˜π‘€) / (π΅β€˜π‘)) Β· (𝑧↑𝐷)))
4948ply1term 26154 . . . . . 6 ((𝑆 βŠ† β„‚ ∧ ((π΄β€˜π‘€) / (π΅β€˜π‘)) ∈ 𝑆 ∧ 𝐷 ∈ β„•0) β†’ 𝐻 ∈ (Polyβ€˜π‘†))
503, 46, 47, 49syl3anc 1368 . . . . 5 (πœ‘ β†’ 𝐻 ∈ (Polyβ€˜π‘†))
5150adantr 479 . . . 4 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ 𝐻 ∈ (Polyβ€˜π‘†))
52 simpr 483 . . . 4 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ 𝑝 ∈ (Polyβ€˜π‘†))
534adantlr 713 . . . 4 (((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)
5451, 52, 53plyadd 26167 . . 3 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ (𝐻 ∘f + 𝑝) ∈ (Polyβ€˜π‘†))
55 cnex 11217 . . . . . . . . 9 β„‚ ∈ V
5655a1i 11 . . . . . . . 8 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ β„‚ ∈ V)
571adantr 479 . . . . . . . . 9 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ 𝐹 ∈ (Polyβ€˜π‘†))
58 plyf 26148 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝐹:β„‚βŸΆβ„‚)
5957, 58syl 17 . . . . . . . 8 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ 𝐹:β„‚βŸΆβ„‚)
60 mulcl 11220 . . . . . . . . . 10 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) β†’ (π‘₯ Β· 𝑦) ∈ β„‚)
6160adantl 480 . . . . . . . . 9 (((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ (π‘₯ Β· 𝑦) ∈ β„‚)
62 plyf 26148 . . . . . . . . . 10 (𝐻 ∈ (Polyβ€˜π‘†) β†’ 𝐻:β„‚βŸΆβ„‚)
6351, 62syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ 𝐻:β„‚βŸΆβ„‚)
6418adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ 𝐺 ∈ (Polyβ€˜π‘†))
65 plyf 26148 . . . . . . . . . 10 (𝐺 ∈ (Polyβ€˜π‘†) β†’ 𝐺:β„‚βŸΆβ„‚)
6664, 65syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ 𝐺:β„‚βŸΆβ„‚)
67 inidm 4213 . . . . . . . . 9 (β„‚ ∩ β„‚) = β„‚
6861, 63, 66, 56, 56, 67off 7699 . . . . . . . 8 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ (𝐻 ∘f Β· 𝐺):β„‚βŸΆβ„‚)
69 plyf 26148 . . . . . . . . . 10 (𝑝 ∈ (Polyβ€˜π‘†) β†’ 𝑝:β„‚βŸΆβ„‚)
7069adantl 480 . . . . . . . . 9 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ 𝑝:β„‚βŸΆβ„‚)
7161, 66, 70, 56, 56, 67off 7699 . . . . . . . 8 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ (𝐺 ∘f Β· 𝑝):β„‚βŸΆβ„‚)
72 subsub4 11521 . . . . . . . . 9 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ ((π‘₯ βˆ’ 𝑦) βˆ’ 𝑧) = (π‘₯ βˆ’ (𝑦 + 𝑧)))
7372adantl 480 . . . . . . . 8 (((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚)) β†’ ((π‘₯ βˆ’ 𝑦) βˆ’ 𝑧) = (π‘₯ βˆ’ (𝑦 + 𝑧)))
7456, 59, 68, 71, 73caofass 7719 . . . . . . 7 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ ((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = (𝐹 ∘f βˆ’ ((𝐻 ∘f Β· 𝐺) ∘f + (𝐺 ∘f Β· 𝑝))))
75 mulcom 11222 . . . . . . . . . . . 12 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))
7675adantl 480 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))
7756, 63, 66, 76caofcom 7717 . . . . . . . . . 10 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ (𝐻 ∘f Β· 𝐺) = (𝐺 ∘f Β· 𝐻))
7877oveq1d 7430 . . . . . . . . 9 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ ((𝐻 ∘f Β· 𝐺) ∘f + (𝐺 ∘f Β· 𝑝)) = ((𝐺 ∘f Β· 𝐻) ∘f + (𝐺 ∘f Β· 𝑝)))
79 adddi 11225 . . . . . . . . . . 11 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))
8079adantl 480 . . . . . . . . . 10 (((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚)) β†’ (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))
8156, 66, 63, 70, 80caofdi 7721 . . . . . . . . 9 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)) = ((𝐺 ∘f Β· 𝐻) ∘f + (𝐺 ∘f Β· 𝑝)))
8278, 81eqtr4d 2768 . . . . . . . 8 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ ((𝐻 ∘f Β· 𝐺) ∘f + (𝐺 ∘f Β· 𝑝)) = (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))
8382oveq2d 7431 . . . . . . 7 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ (𝐹 ∘f βˆ’ ((𝐻 ∘f Β· 𝐺) ∘f + (𝐺 ∘f Β· 𝑝))) = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))))
8474, 83eqtrd 2765 . . . . . 6 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ ((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))))
8584eqeq1d 2727 . . . . 5 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝 ↔ (𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))) = 0𝑝))
8684fveq2d 6895 . . . . . 6 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) = (degβ€˜(𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))))
8786breq1d 5153 . . . . 5 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ ((degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁 ↔ (degβ€˜(𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))) < 𝑁))
8885, 87orbi12d 916 . . . 4 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) β†’ ((((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝 ∨ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁) ↔ ((𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))) = 0𝑝 ∨ (degβ€˜(𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))) < 𝑁)))
8988biimpa 475 . . 3 (((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) ∧ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝 ∨ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁)) β†’ ((𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))) = 0𝑝 ∨ (degβ€˜(𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))) < 𝑁))
90 plydiv.r . . . . . . 7 𝑅 = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· π‘ž))
91 oveq2 7423 . . . . . . . 8 (π‘ž = (𝐻 ∘f + 𝑝) β†’ (𝐺 ∘f Β· π‘ž) = (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))
9291oveq2d 7431 . . . . . . 7 (π‘ž = (𝐻 ∘f + 𝑝) β†’ (𝐹 ∘f βˆ’ (𝐺 ∘f Β· π‘ž)) = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))))
9390, 92eqtrid 2777 . . . . . 6 (π‘ž = (𝐻 ∘f + 𝑝) β†’ 𝑅 = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))))
9493eqeq1d 2727 . . . . 5 (π‘ž = (𝐻 ∘f + 𝑝) β†’ (𝑅 = 0𝑝 ↔ (𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))) = 0𝑝))
9593fveq2d 6895 . . . . . 6 (π‘ž = (𝐻 ∘f + 𝑝) β†’ (degβ€˜π‘…) = (degβ€˜(𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))))
9695breq1d 5153 . . . . 5 (π‘ž = (𝐻 ∘f + 𝑝) β†’ ((degβ€˜π‘…) < 𝑁 ↔ (degβ€˜(𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))) < 𝑁))
9794, 96orbi12d 916 . . . 4 (π‘ž = (𝐻 ∘f + 𝑝) β†’ ((𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < 𝑁) ↔ ((𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))) = 0𝑝 ∨ (degβ€˜(𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))) < 𝑁)))
9897rspcev 3602 . . 3 (((𝐻 ∘f + 𝑝) ∈ (Polyβ€˜π‘†) ∧ ((𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝))) = 0𝑝 ∨ (degβ€˜(𝐹 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 ∘f + 𝑝)))) < 𝑁)) β†’ βˆƒπ‘ž ∈ (Polyβ€˜π‘†)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < 𝑁))
9954, 89, 98syl2an2r 683 . 2 (((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π‘†)) ∧ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝 ∨ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁)) β†’ βˆƒπ‘ž ∈ (Polyβ€˜π‘†)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < 𝑁))
10050, 18, 4, 5plymul 26168 . . . . . 6 (πœ‘ β†’ (𝐻 ∘f Β· 𝐺) ∈ (Polyβ€˜π‘†))
101 eqid 2725 . . . . . . 7 (degβ€˜(𝐻 ∘f Β· 𝐺)) = (degβ€˜(𝐻 ∘f Β· 𝐺))
10212, 101dgrsub 26223 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ (𝐻 ∘f Β· 𝐺) ∈ (Polyβ€˜π‘†)) β†’ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) ≀ if(𝑀 ≀ (degβ€˜(𝐻 ∘f Β· 𝐺)), (degβ€˜(𝐻 ∘f Β· 𝐺)), 𝑀))
1031, 100, 102syl2anc 582 . . . . 5 (πœ‘ β†’ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) ≀ if(𝑀 ≀ (degβ€˜(𝐻 ∘f Β· 𝐺)), (degβ€˜(𝐻 ∘f Β· 𝐺)), 𝑀))
104 plydiv.fz . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐹 β‰  0𝑝)
10512, 9dgreq0 26216 . . . . . . . . . . . . . . 15 (𝐹 ∈ (Polyβ€˜π‘†) β†’ (𝐹 = 0𝑝 ↔ (π΄β€˜π‘€) = 0))
1061, 105syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝐹 = 0𝑝 ↔ (π΄β€˜π‘€) = 0))
107106necon3bid 2975 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝐹 β‰  0𝑝 ↔ (π΄β€˜π‘€) β‰  0))
108104, 107mpbid 231 . . . . . . . . . . . 12 (πœ‘ β†’ (π΄β€˜π‘€) β‰  0)
10917, 27, 108, 32divne0d 12034 . . . . . . . . . . 11 (πœ‘ β†’ ((π΄β€˜π‘€) / (π΅β€˜π‘)) β‰  0)
1103, 46sseldd 3973 . . . . . . . . . . . . 13 (πœ‘ β†’ ((π΄β€˜π‘€) / (π΅β€˜π‘)) ∈ β„‚)
11148coe1term 26209 . . . . . . . . . . . . 13 ((((π΄β€˜π‘€) / (π΅β€˜π‘)) ∈ β„‚ ∧ 𝐷 ∈ β„•0 ∧ 𝐷 ∈ β„•0) β†’ ((coeffβ€˜π»)β€˜π·) = if(𝐷 = 𝐷, ((π΄β€˜π‘€) / (π΅β€˜π‘)), 0))
112110, 47, 47, 111syl3anc 1368 . . . . . . . . . . . 12 (πœ‘ β†’ ((coeffβ€˜π»)β€˜π·) = if(𝐷 = 𝐷, ((π΄β€˜π‘€) / (π΅β€˜π‘)), 0))
113 eqid 2725 . . . . . . . . . . . . 13 𝐷 = 𝐷
114113iftruei 4531 . . . . . . . . . . . 12 if(𝐷 = 𝐷, ((π΄β€˜π‘€) / (π΅β€˜π‘)), 0) = ((π΄β€˜π‘€) / (π΅β€˜π‘))
115112, 114eqtrdi 2781 . . . . . . . . . . 11 (πœ‘ β†’ ((coeffβ€˜π»)β€˜π·) = ((π΄β€˜π‘€) / (π΅β€˜π‘)))
116 c0ex 11236 . . . . . . . . . . . . 13 0 ∈ V
117116fvconst2 7211 . . . . . . . . . . . 12 (𝐷 ∈ β„•0 β†’ ((β„•0 Γ— {0})β€˜π·) = 0)
11847, 117syl 17 . . . . . . . . . . 11 (πœ‘ β†’ ((β„•0 Γ— {0})β€˜π·) = 0)
119109, 115, 1183netr4d 3008 . . . . . . . . . 10 (πœ‘ β†’ ((coeffβ€˜π»)β€˜π·) β‰  ((β„•0 Γ— {0})β€˜π·))
120 fveq2 6891 . . . . . . . . . . . . 13 (𝐻 = 0𝑝 β†’ (coeffβ€˜π») = (coeffβ€˜0𝑝))
121 coe0 26206 . . . . . . . . . . . . 13 (coeffβ€˜0𝑝) = (β„•0 Γ— {0})
122120, 121eqtrdi 2781 . . . . . . . . . . . 12 (𝐻 = 0𝑝 β†’ (coeffβ€˜π») = (β„•0 Γ— {0}))
123122fveq1d 6893 . . . . . . . . . . 11 (𝐻 = 0𝑝 β†’ ((coeffβ€˜π»)β€˜π·) = ((β„•0 Γ— {0})β€˜π·))
124123necon3i 2963 . . . . . . . . . 10 (((coeffβ€˜π»)β€˜π·) β‰  ((β„•0 Γ— {0})β€˜π·) β†’ 𝐻 β‰  0𝑝)
125119, 124syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐻 β‰  0𝑝)
126 eqid 2725 . . . . . . . . . 10 (degβ€˜π») = (degβ€˜π»)
127126, 22dgrmul 26221 . . . . . . . . 9 (((𝐻 ∈ (Polyβ€˜π‘†) ∧ 𝐻 β‰  0𝑝) ∧ (𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝)) β†’ (degβ€˜(𝐻 ∘f Β· 𝐺)) = ((degβ€˜π») + 𝑁))
12850, 125, 18, 28, 127syl22anc 837 . . . . . . . 8 (πœ‘ β†’ (degβ€˜(𝐻 ∘f Β· 𝐺)) = ((degβ€˜π») + 𝑁))
12948dgr1term 26210 . . . . . . . . . . . 12 ((((π΄β€˜π‘€) / (π΅β€˜π‘)) ∈ β„‚ ∧ ((π΄β€˜π‘€) / (π΅β€˜π‘)) β‰  0 ∧ 𝐷 ∈ β„•0) β†’ (degβ€˜π») = 𝐷)
130110, 109, 47, 129syl3anc 1368 . . . . . . . . . . 11 (πœ‘ β†’ (degβ€˜π») = 𝐷)
131 plydiv.e . . . . . . . . . . 11 (πœ‘ β†’ (𝑀 βˆ’ 𝑁) = 𝐷)
132130, 131eqtr4d 2768 . . . . . . . . . 10 (πœ‘ β†’ (degβ€˜π») = (𝑀 βˆ’ 𝑁))
133132oveq1d 7430 . . . . . . . . 9 (πœ‘ β†’ ((degβ€˜π») + 𝑁) = ((𝑀 βˆ’ 𝑁) + 𝑁))
13415nn0cnd 12562 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 ∈ β„‚)
13525nn0cnd 12562 . . . . . . . . . 10 (πœ‘ β†’ 𝑁 ∈ β„‚)
136134, 135npcand 11603 . . . . . . . . 9 (πœ‘ β†’ ((𝑀 βˆ’ 𝑁) + 𝑁) = 𝑀)
137133, 136eqtrd 2765 . . . . . . . 8 (πœ‘ β†’ ((degβ€˜π») + 𝑁) = 𝑀)
138128, 137eqtrd 2765 . . . . . . 7 (πœ‘ β†’ (degβ€˜(𝐻 ∘f Β· 𝐺)) = 𝑀)
139138ifeq1d 4543 . . . . . 6 (πœ‘ β†’ if(𝑀 ≀ (degβ€˜(𝐻 ∘f Β· 𝐺)), (degβ€˜(𝐻 ∘f Β· 𝐺)), 𝑀) = if(𝑀 ≀ (degβ€˜(𝐻 ∘f Β· 𝐺)), 𝑀, 𝑀))
140 ifid 4564 . . . . . 6 if(𝑀 ≀ (degβ€˜(𝐻 ∘f Β· 𝐺)), 𝑀, 𝑀) = 𝑀
141139, 140eqtrdi 2781 . . . . 5 (πœ‘ β†’ if(𝑀 ≀ (degβ€˜(𝐻 ∘f Β· 𝐺)), (degβ€˜(𝐻 ∘f Β· 𝐺)), 𝑀) = 𝑀)
142103, 141breqtrd 5169 . . . 4 (πœ‘ β†’ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) ≀ 𝑀)
143 eqid 2725 . . . . . . . 8 (coeffβ€˜(𝐻 ∘f Β· 𝐺)) = (coeffβ€˜(𝐻 ∘f Β· 𝐺))
1449, 143coesub 26207 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ (𝐻 ∘f Β· 𝐺) ∈ (Polyβ€˜π‘†)) β†’ (coeffβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) = (𝐴 ∘f βˆ’ (coeffβ€˜(𝐻 ∘f Β· 𝐺))))
1451, 100, 144syl2anc 582 . . . . . 6 (πœ‘ β†’ (coeffβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) = (𝐴 ∘f βˆ’ (coeffβ€˜(𝐻 ∘f Β· 𝐺))))
146145fveq1d 6893 . . . . 5 (πœ‘ β†’ ((coeffβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)))β€˜π‘€) = ((𝐴 ∘f βˆ’ (coeffβ€˜(𝐻 ∘f Β· 𝐺)))β€˜π‘€))
1479coef3 26182 . . . . . . . 8 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝐴:β„•0βŸΆβ„‚)
148 ffn 6716 . . . . . . . 8 (𝐴:β„•0βŸΆβ„‚ β†’ 𝐴 Fn β„•0)
1491, 147, 1483syl 18 . . . . . . 7 (πœ‘ β†’ 𝐴 Fn β„•0)
150143coef3 26182 . . . . . . . 8 ((𝐻 ∘f Β· 𝐺) ∈ (Polyβ€˜π‘†) β†’ (coeffβ€˜(𝐻 ∘f Β· 𝐺)):β„•0βŸΆβ„‚)
151 ffn 6716 . . . . . . . 8 ((coeffβ€˜(𝐻 ∘f Β· 𝐺)):β„•0βŸΆβ„‚ β†’ (coeffβ€˜(𝐻 ∘f Β· 𝐺)) Fn β„•0)
152100, 150, 1513syl 18 . . . . . . 7 (πœ‘ β†’ (coeffβ€˜(𝐻 ∘f Β· 𝐺)) Fn β„•0)
153 nn0ex 12506 . . . . . . . 8 β„•0 ∈ V
154153a1i 11 . . . . . . 7 (πœ‘ β†’ β„•0 ∈ V)
155 inidm 4213 . . . . . . 7 (β„•0 ∩ β„•0) = β„•0
156 eqidd 2726 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ β„•0) β†’ (π΄β€˜π‘€) = (π΄β€˜π‘€))
157 eqid 2725 . . . . . . . . . . 11 (coeffβ€˜π») = (coeffβ€˜π»)
158157, 19, 126, 22coemulhi 26204 . . . . . . . . . 10 ((𝐻 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ ((coeffβ€˜(𝐻 ∘f Β· 𝐺))β€˜((degβ€˜π») + 𝑁)) = (((coeffβ€˜π»)β€˜(degβ€˜π»)) Β· (π΅β€˜π‘)))
15950, 18, 158syl2anc 582 . . . . . . . . 9 (πœ‘ β†’ ((coeffβ€˜(𝐻 ∘f Β· 𝐺))β€˜((degβ€˜π») + 𝑁)) = (((coeffβ€˜π»)β€˜(degβ€˜π»)) Β· (π΅β€˜π‘)))
160137fveq2d 6895 . . . . . . . . 9 (πœ‘ β†’ ((coeffβ€˜(𝐻 ∘f Β· 𝐺))β€˜((degβ€˜π») + 𝑁)) = ((coeffβ€˜(𝐻 ∘f Β· 𝐺))β€˜π‘€))
161130fveq2d 6895 . . . . . . . . . . . 12 (πœ‘ β†’ ((coeffβ€˜π»)β€˜(degβ€˜π»)) = ((coeffβ€˜π»)β€˜π·))
162161, 115eqtrd 2765 . . . . . . . . . . 11 (πœ‘ β†’ ((coeffβ€˜π»)β€˜(degβ€˜π»)) = ((π΄β€˜π‘€) / (π΅β€˜π‘)))
163162oveq1d 7430 . . . . . . . . . 10 (πœ‘ β†’ (((coeffβ€˜π»)β€˜(degβ€˜π»)) Β· (π΅β€˜π‘)) = (((π΄β€˜π‘€) / (π΅β€˜π‘)) Β· (π΅β€˜π‘)))
16417, 27, 32divcan1d 12019 . . . . . . . . . 10 (πœ‘ β†’ (((π΄β€˜π‘€) / (π΅β€˜π‘)) Β· (π΅β€˜π‘)) = (π΄β€˜π‘€))
165163, 164eqtrd 2765 . . . . . . . . 9 (πœ‘ β†’ (((coeffβ€˜π»)β€˜(degβ€˜π»)) Β· (π΅β€˜π‘)) = (π΄β€˜π‘€))
166159, 160, 1653eqtr3d 2773 . . . . . . . 8 (πœ‘ β†’ ((coeffβ€˜(𝐻 ∘f Β· 𝐺))β€˜π‘€) = (π΄β€˜π‘€))
167166adantr 479 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ β„•0) β†’ ((coeffβ€˜(𝐻 ∘f Β· 𝐺))β€˜π‘€) = (π΄β€˜π‘€))
168149, 152, 154, 154, 155, 156, 167ofval 7692 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ β„•0) β†’ ((𝐴 ∘f βˆ’ (coeffβ€˜(𝐻 ∘f Β· 𝐺)))β€˜π‘€) = ((π΄β€˜π‘€) βˆ’ (π΄β€˜π‘€)))
16915, 168mpdan 685 . . . . 5 (πœ‘ β†’ ((𝐴 ∘f βˆ’ (coeffβ€˜(𝐻 ∘f Β· 𝐺)))β€˜π‘€) = ((π΄β€˜π‘€) βˆ’ (π΄β€˜π‘€)))
17017subidd 11587 . . . . 5 (πœ‘ β†’ ((π΄β€˜π‘€) βˆ’ (π΄β€˜π‘€)) = 0)
171146, 169, 1703eqtrd 2769 . . . 4 (πœ‘ β†’ ((coeffβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)))β€˜π‘€) = 0)
1721, 100, 4, 5, 7plysub 26169 . . . . . . . . . 10 (πœ‘ β†’ (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∈ (Polyβ€˜π‘†))
173 dgrcl 26183 . . . . . . . . . 10 ((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∈ (Polyβ€˜π‘†) β†’ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) ∈ β„•0)
174172, 173syl 17 . . . . . . . . 9 (πœ‘ β†’ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) ∈ β„•0)
175174nn0red 12561 . . . . . . . 8 (πœ‘ β†’ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) ∈ ℝ)
17615nn0red 12561 . . . . . . . 8 (πœ‘ β†’ 𝑀 ∈ ℝ)
17725nn0red 12561 . . . . . . . 8 (πœ‘ β†’ 𝑁 ∈ ℝ)
178175, 176, 177ltsub1d 11851 . . . . . . 7 (πœ‘ β†’ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) < 𝑀 ↔ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < (𝑀 βˆ’ 𝑁)))
179131breq2d 5155 . . . . . . 7 (πœ‘ β†’ (((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < (𝑀 βˆ’ 𝑁) ↔ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < 𝐷))
180178, 179bitrd 278 . . . . . 6 (πœ‘ β†’ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) < 𝑀 ↔ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < 𝐷))
181180orbi2d 913 . . . . 5 (πœ‘ β†’ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝 ∨ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) < 𝑀) ↔ ((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝 ∨ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < 𝐷)))
182 eqid 2725 . . . . . . 7 (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) = (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)))
183 eqid 2725 . . . . . . 7 (coeffβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) = (coeffβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)))
184182, 183dgrlt 26217 . . . . . 6 (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∈ (Polyβ€˜π‘†) ∧ 𝑀 ∈ β„•0) β†’ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝 ∨ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) < 𝑀) ↔ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) ≀ 𝑀 ∧ ((coeffβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)))β€˜π‘€) = 0)))
185172, 15, 184syl2anc 582 . . . . 5 (πœ‘ β†’ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝 ∨ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) < 𝑀) ↔ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) ≀ 𝑀 ∧ ((coeffβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)))β€˜π‘€) = 0)))
186181, 185bitr3d 280 . . . 4 (πœ‘ β†’ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝 ∨ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < 𝐷) ↔ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) ≀ 𝑀 ∧ ((coeffβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)))β€˜π‘€) = 0)))
187142, 171, 186mpbir2and 711 . . 3 (πœ‘ β†’ ((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝 ∨ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < 𝐷))
188 eqeq1 2729 . . . . . 6 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ (𝑓 = 0𝑝 ↔ (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝))
189 fveq2 6891 . . . . . . . 8 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ (degβ€˜π‘“) = (degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))))
190189oveq1d 7430 . . . . . . 7 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ ((degβ€˜π‘“) βˆ’ 𝑁) = ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁))
191190breq1d 5153 . . . . . 6 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ (((degβ€˜π‘“) βˆ’ 𝑁) < 𝐷 ↔ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < 𝐷))
192188, 191orbi12d 916 . . . . 5 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ ((𝑓 = 0𝑝 ∨ ((degβ€˜π‘“) βˆ’ 𝑁) < 𝐷) ↔ ((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝 ∨ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < 𝐷)))
193 plydiv.u . . . . . . . . 9 π‘ˆ = (𝑓 ∘f βˆ’ (𝐺 ∘f Β· 𝑝))
194 oveq1 7422 . . . . . . . . 9 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ (𝑓 ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = ((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)))
195193, 194eqtrid 2777 . . . . . . . 8 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ π‘ˆ = ((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)))
196195eqeq1d 2727 . . . . . . 7 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ (π‘ˆ = 0𝑝 ↔ ((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝))
197195fveq2d 6895 . . . . . . . 8 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ (degβ€˜π‘ˆ) = (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))))
198197breq1d 5153 . . . . . . 7 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ ((degβ€˜π‘ˆ) < 𝑁 ↔ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁))
199196, 198orbi12d 916 . . . . . 6 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ ((π‘ˆ = 0𝑝 ∨ (degβ€˜π‘ˆ) < 𝑁) ↔ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝 ∨ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁)))
200199rexbidv 3169 . . . . 5 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π‘†)(π‘ˆ = 0𝑝 ∨ (degβ€˜π‘ˆ) < 𝑁) ↔ βˆƒπ‘ ∈ (Polyβ€˜π‘†)(((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝 ∨ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁)))
201192, 200imbi12d 343 . . . 4 (𝑓 = (𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) β†’ (((𝑓 = 0𝑝 ∨ ((degβ€˜π‘“) βˆ’ 𝑁) < 𝐷) β†’ βˆƒπ‘ ∈ (Polyβ€˜π‘†)(π‘ˆ = 0𝑝 ∨ (degβ€˜π‘ˆ) < 𝑁)) ↔ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝 ∨ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < 𝐷) β†’ βˆƒπ‘ ∈ (Polyβ€˜π‘†)(((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝 ∨ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁))))
202 plydiv.al . . . 4 (πœ‘ β†’ βˆ€π‘“ ∈ (Polyβ€˜π‘†)((𝑓 = 0𝑝 ∨ ((degβ€˜π‘“) βˆ’ 𝑁) < 𝐷) β†’ βˆƒπ‘ ∈ (Polyβ€˜π‘†)(π‘ˆ = 0𝑝 ∨ (degβ€˜π‘ˆ) < 𝑁)))
203201, 202, 172rspcdva 3603 . . 3 (πœ‘ β†’ (((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) = 0𝑝 ∨ ((degβ€˜(𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺))) βˆ’ 𝑁) < 𝐷) β†’ βˆƒπ‘ ∈ (Polyβ€˜π‘†)(((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝 ∨ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁)))
204187, 203mpd 15 . 2 (πœ‘ β†’ βˆƒπ‘ ∈ (Polyβ€˜π‘†)(((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝)) = 0𝑝 ∨ (degβ€˜((𝐹 ∘f βˆ’ (𝐻 ∘f Β· 𝐺)) ∘f βˆ’ (𝐺 ∘f Β· 𝑝))) < 𝑁))
20599, 204r19.29a 3152 1 (πœ‘ β†’ βˆƒπ‘ž ∈ (Polyβ€˜π‘†)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < 𝑁))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  βˆƒwrex 3060  Vcvv 3463   βŠ† wss 3940  ifcif 4524  {csn 4624   class class class wbr 5143   ↦ cmpt 5226   Γ— cxp 5670   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7415   ∘f cof 7679  β„‚cc 11134  0cc0 11136  1c1 11137   + caddc 11139   Β· cmul 11141   < clt 11276   ≀ cle 11277   βˆ’ cmin 11472  -cneg 11473   / cdiv 11899  β„•0cn0 12500  β†‘cexp 14056  0𝑝c0p 25614  Polycply 26134  coeffccoe 26136  degcdgr 26137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-inf2 9662  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-pre-sup 11214
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-isom 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7681  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-map 8843  df-pm 8844  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-sup 9463  df-inf 9464  df-oi 9531  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-n0 12501  df-z 12587  df-uz 12851  df-rp 13005  df-fz 13515  df-fzo 13658  df-fl 13787  df-seq 13997  df-exp 14057  df-hash 14320  df-cj 15076  df-re 15077  df-im 15078  df-sqrt 15212  df-abs 15213  df-clim 15462  df-rlim 15463  df-sum 15663  df-0p 25615  df-ply 26138  df-coe 26140  df-dgr 26141
This theorem is referenced by:  plydivex  26248
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