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Theorem diophun 41501
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 41492 . . 3 (𝐴 ∈ (Diophβ€˜π‘) β†’ 𝑁 ∈ β„•0)
2 nnex 12217 . . . . . 6 β„• ∈ V
32jctr 525 . . . . 5 (𝑁 ∈ β„•0 β†’ (𝑁 ∈ β„•0 ∧ β„• ∈ V))
4 1z 12591 . . . . . . 7 1 ∈ β„€
5 nnuz 12864 . . . . . . . 8 β„• = (β„€β‰₯β€˜1)
65uzinf 13929 . . . . . . 7 (1 ∈ β„€ β†’ Β¬ β„• ∈ Fin)
74, 6ax-mp 5 . . . . . 6 Β¬ β„• ∈ Fin
8 elfznn 13529 . . . . . . 7 (π‘Ž ∈ (1...𝑁) β†’ π‘Ž ∈ β„•)
98ssriv 3986 . . . . . 6 (1...𝑁) βŠ† β„•
107, 9pm3.2i 471 . . . . 5 (Β¬ β„• ∈ Fin ∧ (1...𝑁) βŠ† β„•)
11 eldioph2b 41491 . . . . . 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 41491 . . . . . 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 631 . . . . 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 586 . . . 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 3226 . . . . 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 4298 . . . . . . . . 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 3122 . . . . . . . . . . 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 1006 . . . . . . . . . . . . 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 12566 . . . . . . . . . . . . . . . . . . . 20 β„€ ∈ V
20 nn0ssz 12580 . . . . . . . . . . . . . . . . . . . 20 β„•0 βŠ† β„€
21 mapss 8882 . . . . . . . . . . . . . . . . . . . 20 ((β„€ ∈ V ∧ β„•0 βŠ† β„€) β†’ (β„•0 ↑m β„•) βŠ† (β„€ ↑m β„•))
2219, 20, 21mp2an 690 . . . . . . . . . . . . . . . . . . 19 (β„•0 ↑m β„•) βŠ† (β„€ ↑m β„•)
2322sseli 3978 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (β„•0 ↑m β„•) β†’ 𝑑 ∈ (β„€ ↑m β„•))
2423adantl 482 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ 𝑑 ∈ (β„€ ↑m β„•))
25 fveq2 6891 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 β†’ (π‘Žβ€˜π‘’) = (π‘Žβ€˜π‘‘))
26 fveq2 6891 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑑 β†’ (π‘β€˜π‘’) = (π‘β€˜π‘‘))
2725, 26oveq12d 7426 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝑑 β†’ ((π‘Žβ€˜π‘’) Β· (π‘β€˜π‘’)) = ((π‘Žβ€˜π‘‘) Β· (π‘β€˜π‘‘)))
28 eqid 2732 . . . . . . . . . . . . . . . . . 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 7441 . . . . . . . . . . . . . . . . . 18 ((π‘Žβ€˜π‘‘) Β· (π‘β€˜π‘‘)) ∈ V
3027, 28, 29fvmpt 6998 . . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ π‘Ž ∈ (mzPolyβ€˜β„•))
34 mzpf 41464 . . . . . . . . . . . . . . . . . . 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 7087 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (π‘Žβ€˜π‘‘) ∈ β„€)
3736zcnd 12666 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (π‘Žβ€˜π‘‘) ∈ β„‚)
38 simplrr 776 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ 𝑐 ∈ (mzPolyβ€˜β„•))
39 mzpf 41464 . . . . . . . . . . . . . . . . . . 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 7087 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (π‘β€˜π‘‘) ∈ β„€)
4241zcnd 12666 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) ∧ 𝑑 ∈ (β„•0 ↑m β„•)) β†’ (π‘β€˜π‘‘) ∈ β„‚)
4337, 42mul0ord 11863 . . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . 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 284 . . . . . . . . . . . 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 3176 . . . . . . . . . . 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 284 . . . . . . . . . 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 2801 . . . . . . . . 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 2784 . . . . . . . 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 483 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ 𝑁 ∈ β„•0)
522, 9pm3.2i 471 . . . . . . . . . 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 769 . . . . . . . . . . . . 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 6960 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ π‘Ž = (𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘Žβ€˜π‘’)))
5756, 54eqeltrrd 2834 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ (𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘Žβ€˜π‘’)) ∈ (mzPolyβ€˜β„•))
58 simprr 771 . . . . . . . . . . . . 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 6960 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ 𝑐 = (𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘β€˜π‘’)))
6160, 58eqeltrrd 2834 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ (π‘Ž ∈ (mzPolyβ€˜β„•) ∧ 𝑐 ∈ (mzPolyβ€˜β„•))) β†’ (𝑒 ∈ (β„€ ↑m β„•) ↦ (π‘β€˜π‘’)) ∈ (mzPolyβ€˜β„•))
62 mzpmulmpt 41470 . . . . . . . . . 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 584 . . . . . . . . 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 41490 . . . . . . . . 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 1371 . . . . . . . 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 2833 . . . . . . 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 4158 . . . . . . . 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 2818 . . . . . . 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 3211 . . . . 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 667 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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆƒwrex 3070  Vcvv 3474   βˆͺ cun 3946   βŠ† wss 3948   ↦ cmpt 5231   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819  Fincfn 8938  0cc0 11109  1c1 11110   Β· cmul 11114  β„•cn 12211  β„•0cn0 12471  β„€cz 12557  ...cfz 13483  mzPolycmzp 41450  Diophcdioph 41483
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 7724  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
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 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-oadd 8469  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-dju 9895  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-hash 14290  df-mzpcl 41451  df-mzp 41452  df-dioph 41484
This theorem is referenced by:  orrabdioph  41509
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