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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diophun Structured version   Visualization version   GIF version
Theorem diophun 41559
Description: If two sets are Diophantine, so is their union. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Assertion
Ref Expression
diophun ((𝐴 ∈ (Diophβ€˜π‘) ∧ 𝐡 ∈ (Diophβ€˜π‘)) β†’ (𝐴 βˆͺ 𝐡) ∈ (Diophβ€˜π‘))
Proof of Theorem diophun
Dummy variables π‘Ž 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldiophelnn0 41550 . . 3 (𝐴 ∈ (Diophβ€˜π‘) β†’ 𝑁 ∈ β„•0)
2 nnex 12218 . . . . . 6 β„• ∈ V
32jctr 526 . . . . 5 (𝑁 ∈ β„•0 β†’ (𝑁 ∈ β„•0 ∧ β„• ∈ V))
4 1z 12592 . . . . . . 7 1 ∈ β„€
5 nnuz 12865 . . . . . . . 8 β„• = (β„€β‰₯β€˜1)
65uzinf 13930 . . . . . . 7 (1 ∈ β„€ β†’ Β¬ β„• ∈ Fin)
74, 6ax-mp 5 . . . . . 6 Β¬ β„• ∈ Fin
8 elfznn 13530 . . . . . . 7 (π‘Ž ∈ (1...𝑁) β†’ π‘Ž ∈ β„•)
98ssriv 3987 . . . . . 6 (1...𝑁) βŠ† β„•
107, 9pm3.2i 472 . . . . 5 (Β¬ β„• ∈ Fin ∧ (1...𝑁) βŠ† β„•)
11 eldioph2b 41549 . . . . . 6 (((𝑁 ∈ β„•0 ∧ β„• ∈ V) ∧ (Β¬ β„• ∈ Fin ∧ (1...𝑁) βŠ† β„•)) β†’ (𝐴 ∈ (Diophβ€˜π‘) ↔ βˆƒπ‘Ž ∈ (mzPolyβ€˜β„•)𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)}))
12 eldioph2b 41549 . . . . . 6 (((𝑁 ∈ β„•0 ∧ β„• ∈ V) ∧ (Β¬ β„• ∈ Fin ∧ (1...𝑁) βŠ† β„•)) β†’ (𝐡 ∈ (Diophβ€˜π‘) ↔ βˆƒπ‘ ∈ (mzPolyβ€˜β„•)𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}))
1311, 12anbi12d 632 . . . . 5 (((𝑁 ∈ β„•0 ∧ β„• ∈ V) ∧ (Β¬ β„• ∈ Fin ∧ (1...𝑁) βŠ† β„•)) β†’ ((𝐴 ∈ (Diophβ€˜π‘) ∧ 𝐡 ∈ (Diophβ€˜π‘)) ↔ (βˆƒπ‘Ž ∈ (mzPolyβ€˜β„•)𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} ∧ βˆƒπ‘ ∈ (mzPolyβ€˜β„•)𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)})))
143, 10, 13sylancl 587 . . . 4 (𝑁 ∈ β„•0 β†’ ((𝐴 ∈ (Diophβ€˜π‘) ∧ 𝐡 ∈ (Diophβ€˜π‘)) ↔ (βˆƒπ‘Ž ∈ (mzPolyβ€˜β„•)𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} ∧ βˆƒπ‘ ∈ (mzPolyβ€˜β„•)𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)})))
15 reeanv 3227 . . . . 5 (βˆƒπ‘Ž ∈ (mzPolyβ€˜β„•)βˆƒπ‘ ∈ (mzPolyβ€˜β„•)(𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} ∧ 𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) ↔ (βˆƒπ‘Ž ∈ (mzPolyβ€˜β„•)𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} ∧ βˆƒπ‘ ∈ (mzPolyβ€˜β„•)𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}))
16 unab 4299 . . . . . . . . 9 ({𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} βˆͺ {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) = {𝑏 ∣ (βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0) ∨ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0))}
17 r19.43 3123 . . . . . . . . . . 11 (βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)((𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0) ∨ (𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)) ↔ (βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0) ∨ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)))
18 andi 1007 . . . . . . . . . . . . 13 ((𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((π‘Žβ€˜π‘‘) = 0 ∨ (π‘β€˜π‘‘) = 0)) ↔ ((𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0) ∨ (𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)))
19 zex 12567 . . . . . . . . . . . . . . . . . . . 20 β„€ ∈ V
20 nn0ssz 12581 . . . . . . . . . . . . . . . . . . . 20 β„•0 βŠ† β„€
21 mapss 8883 . . . . . . . . . . . . . . . . . . . 20 ((β„€ ∈ V ∧ β„•0 βŠ† β„€) β†’ (β„•0 ↑m β„•) βŠ† (β„€ ↑m β„•))
2219, 20, 21mp2an 691 . . . . . . . . . . . . . . . . . . 19 (β„•0 ↑m β„•) βŠ† (β„€ ↑m β„•)
2322sseli 3979 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (β„•0 ↑m β„•) β†’ 𝑑 ∈ (β„€ ↑m β„•))
2423adantl 483 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ 𝑑 ∈ (β„€ ↑m β„•))
25 fveq2 6892 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 β†’ (π‘Žβ€˜π‘’) = (π‘Žβ€˜π‘‘))
26 fveq2 6892 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 β†’ (π‘β€˜π‘’) = (π‘β€˜π‘‘))
2725, 26oveq12d 7427 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝑑 β†’ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)) = ((π‘Žβ€˜π‘‘) Β· (π‘β€˜π‘‘)))
28 eqid 2733 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’))) = (𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))
29 ovex 7442 . . . . . . . . . . . . . . . . . 18 ((π‘Žβ€˜π‘‘) Β· (π‘β€˜π‘‘)) ∈ V
3027, 28, 29fvmpt 6999 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ (β„€ ↑m β„•) β†’ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = ((π‘Žβ€˜π‘‘) Β· (π‘β€˜π‘‘)))
3124, 30syl 17 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = ((π‘Žβ€˜π‘‘) Β· (π‘β€˜π‘‘)))
3231eqeq1d 2735 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0 ↔ ((π‘Žβ€˜π‘‘) Β· (π‘β€˜π‘‘)) = 0))
33 simplrl 776 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ π‘Ž ∈ (mzPolyβ€˜β„•))
34 mzpf 41522 . . . . . . . . . . . . . . . . . . 19 (π‘Ž ∈ (mzPolyβ€˜β„•) β†’ π‘Ž:(β„€ ↑m β„•)βŸΆβ„€)
3533, 34syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ π‘Ž:(β„€ ↑m β„•)βŸΆβ„€)
3635, 24ffvelcdmd 7088 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (π‘Žβ€˜π‘‘) ∈ β„€)
3736zcnd 12667 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (π‘Žβ€˜π‘‘) ∈ β„‚)
38 simplrr 777 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ 𝑐 ∈ (mzPolyβ€˜β„•))
39 mzpf 41522 . . . . . . . . . . . . . . . . . . 19 (𝑐 ∈ (mzPolyβ€˜β„•) β†’ 𝑐:(β„€ ↑m β„•)βŸΆβ„€)
4038, 39syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ 𝑐:(β„€ ↑m β„•)βŸΆβ„€)
4140, 24ffvelcdmd 7088 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (π‘β€˜π‘‘) ∈ β„€)
4241zcnd 12667 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (π‘β€˜π‘‘) ∈ β„‚)
4337, 42mul0ord 11864 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (((π‘Žβ€˜π‘‘) Β· (π‘β€˜π‘‘)) = 0 ↔ ((π‘Žβ€˜π‘‘) = 0 ∨ (π‘β€˜π‘‘) = 0)))
4432, 43bitr2d 280 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (((π‘Žβ€˜π‘‘) = 0 ∨ (π‘β€˜π‘‘) = 0) ↔ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0))
4544anbi2d 630 . . . . . . . . . . . . 13 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ ((𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((π‘Žβ€˜π‘‘) = 0 ∨ (π‘β€˜π‘‘) = 0)) ↔ (𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0)))
4618, 45bitr3id 285 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (((𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0) ∨ (𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)) ↔ (𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0)))
4746rexbidva 3177 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ (βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)((𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0) ∨ (𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)) ↔ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0)))
4817, 47bitr3id 285 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ ((βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0) ∨ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)) ↔ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0)))
4948abbidv 2802 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ {𝑏 ∣ (βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0) ∨ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0))} = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0)})
5016, 49eqtrid 2785 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ ({𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} βˆͺ {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0)})
51 simpl 484 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ 𝑁 ∈ β„•0)
522, 9pm3.2i 472 . . . . . . . . . 10 (β„• ∈ V ∧ (1...𝑁) βŠ† β„•)
5352a1i 11 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ (β„• ∈ V ∧ (1...𝑁) βŠ† β„•))
54 simprl 770 . . . . . . . . . . . . 13 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ π‘Ž ∈ (mzPolyβ€˜β„•))
5554, 34syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ π‘Ž:(β„€ ↑m β„•)βŸΆβ„€)
5655feqmptd 6961 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ π‘Ž = (𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘Žβ€˜π‘’)))
5756, 54eqeltrrd 2835 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ (𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘Žβ€˜π‘’)) ∈ (mzPolyβ€˜β„•))
58 simprr 772 . . . . . . . . . . . . 13 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ 𝑐 ∈ (mzPolyβ€˜β„•))
5958, 39syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ 𝑐:(β„€ ↑m β„•)βŸΆβ„€)
6059feqmptd 6961 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ 𝑐 = (𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘β€˜π‘’)))
6160, 58eqeltrrd 2835 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ (𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘β€˜π‘’)) ∈ (mzPolyβ€˜β„•))
62 mzpmulmpt 41528 . . . . . . . . . 10 (((𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘Žβ€˜π‘’)) ∈ (mzPolyβ€˜β„•) ∧ (𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘β€˜π‘’)) ∈ (mzPolyβ€˜β„•)) β†’ (𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’))) ∈ (mzPolyβ€˜β„•))
6357, 61, 62syl2anc 585 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ (𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’))) ∈ (mzPolyβ€˜β„•))
64 eldioph2 41548 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ (β„• ∈ V ∧ (1...𝑁) βŠ† β„•) ∧ (𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’))) ∈ (mzPolyβ€˜β„•)) β†’ {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0)} ∈ (Diophβ€˜π‘))
6551, 53, 63, 64syl3anc 1372 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m β„•) ↦ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)))β€˜π‘‘) = 0)} ∈ (Diophβ€˜π‘))
6650, 65eqeltrd 2834 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ ({𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} βˆͺ {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) ∈ (Diophβ€˜π‘))
67 uneq12 4159 . . . . . . . 8 ((𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} ∧ 𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) β†’ (𝐴 βˆͺ 𝐡) = ({𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} βˆͺ {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}))
6867eleq1d 2819 . . . . . . 7 ((𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} ∧ 𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) β†’ ((𝐴 βˆͺ 𝐡) ∈ (Diophβ€˜π‘) ↔ ({𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} βˆͺ {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) ∈ (Diophβ€˜π‘)))
6966, 68syl5ibrcom 246 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ ((𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} ∧ 𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) β†’ (𝐴 βˆͺ 𝐡) ∈ (Diophβ€˜π‘)))
7069rexlimdvva 3212 . . . . 5 (𝑁 ∈ β„•0 β†’ (βˆƒπ‘Ž ∈ (mzPolyβ€˜β„•)βˆƒπ‘ ∈ (mzPolyβ€˜β„•)(𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} ∧ 𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) β†’ (𝐴 βˆͺ 𝐡) ∈ (Diophβ€˜π‘)))
7115, 70biimtrrid 242 . . . 4 (𝑁 ∈ β„•0 β†’ ((βˆƒπ‘Ž ∈ (mzPolyβ€˜β„•)𝐴 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘Žβ€˜π‘‘) = 0)} ∧ βˆƒπ‘ ∈ (mzPolyβ€˜β„•)𝐡 = {𝑏 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) β†’ (𝐴 βˆͺ 𝐡) ∈ (Diophβ€˜π‘)))
7214, 71sylbid 239 . . 3 (𝑁 ∈ β„•0 β†’ ((𝐴 ∈ (Diophβ€˜π‘) ∧ 𝐡 ∈ (Diophβ€˜π‘)) β†’ (𝐴 βˆͺ 𝐡) ∈ (Diophβ€˜π‘)))
731, 72syl 17 . 2 (𝐴 ∈ (Diophβ€˜π‘) β†’ ((𝐴 ∈ (Diophβ€˜π‘) ∧ 𝐡 ∈ (Diophβ€˜π‘)) β†’ (𝐴 βˆͺ 𝐡) ∈ (Diophβ€˜π‘)))
7473anabsi5 668 1 ((𝐴 ∈ (Diophβ€˜π‘) ∧ 𝐡 ∈ (Diophβ€˜π‘)) β†’ (𝐴 βˆͺ 𝐡) ∈ (Diophβ€˜π‘))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949   ↦ cmpt 5232   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  Fincfn 8939  0cc0 11110  1c1 11111   Β· cmul 11115  β„•cn 12212  β„•0cn0 12472  β„€cz 12558  ...cfz 13484  mzPolycmzp 41508  Diophcdioph 41541
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-inf2 9636  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-of 7670  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-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-mzpcl 41509  df-mzp 41510  df-dioph 41542
This theorem is referenced by:  orrabdioph  41567
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