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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldiophb Structured version   Visualization version   GIF version
Theorem eldiophb 41495
Description: Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Assertion
Ref Expression
eldiophb (𝐷 ∈ (Diophβ€˜π‘) ↔ (𝑁 ∈ β„•0 ∧ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜))𝐷 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
Distinct variable groups:   𝐷,π‘˜,𝑝   π‘˜,𝑁,𝑝,𝑑,𝑒
Allowed substitution hints:   𝐷(𝑒,𝑑)
Proof of Theorem eldiophb
Dummy variables 𝑛 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dioph 41494 . . . 4 Dioph = (𝑛 ∈ β„•0 ↦ ran (π‘˜ ∈ (β„€β‰₯β€˜π‘›), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑛)) ∧ (π‘β€˜π‘’) = 0)}))
21dmmptss 6241 . . 3 dom Dioph βŠ† β„•0
3 elfvdm 6929 . . 3 (𝐷 ∈ (Diophβ€˜π‘) β†’ 𝑁 ∈ dom Dioph)
42, 3sselid 3981 . 2 (𝐷 ∈ (Diophβ€˜π‘) β†’ 𝑁 ∈ β„•0)
5 fveq2 6892 . . . . . . 7 (𝑛 = 𝑁 β†’ (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘))
6 eqidd 2734 . . . . . . 7 (𝑛 = 𝑁 β†’ (mzPolyβ€˜(1...π‘˜)) = (mzPolyβ€˜(1...π‘˜)))
7 oveq2 7417 . . . . . . . . . . . 12 (𝑛 = 𝑁 β†’ (1...𝑛) = (1...𝑁))
87reseq2d 5982 . . . . . . . . . . 11 (𝑛 = 𝑁 β†’ (𝑒 β†Ύ (1...𝑛)) = (𝑒 β†Ύ (1...𝑁)))
98eqeq2d 2744 . . . . . . . . . 10 (𝑛 = 𝑁 β†’ (𝑑 = (𝑒 β†Ύ (1...𝑛)) ↔ 𝑑 = (𝑒 β†Ύ (1...𝑁))))
109anbi1d 631 . . . . . . . . 9 (𝑛 = 𝑁 β†’ ((𝑑 = (𝑒 β†Ύ (1...𝑛)) ∧ (π‘β€˜π‘’) = 0) ↔ (𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
1110rexbidv 3179 . . . . . . . 8 (𝑛 = 𝑁 β†’ (βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑛)) ∧ (π‘β€˜π‘’) = 0) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
1211abbidv 2802 . . . . . . 7 (𝑛 = 𝑁 β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑛)) ∧ (π‘β€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
135, 6, 12mpoeq123dv 7484 . . . . . 6 (𝑛 = 𝑁 β†’ (π‘˜ ∈ (β„€β‰₯β€˜π‘›), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑛)) ∧ (π‘β€˜π‘’) = 0)}) = (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
1413rneqd 5938 . . . . 5 (𝑛 = 𝑁 β†’ ran (π‘˜ ∈ (β„€β‰₯β€˜π‘›), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑛)) ∧ (π‘β€˜π‘’) = 0)}) = ran (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
15 ovex 7442 . . . . . . 7 (β„•0 ↑m (1...𝑁)) ∈ V
1615pwex 5379 . . . . . 6 𝒫 (β„•0 ↑m (1...𝑁)) ∈ V
17 eqid 2733 . . . . . . . 8 (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}) = (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
1817rnmpo 7542 . . . . . . 7 ran (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}) = {𝑑 ∣ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜))𝑑 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}}
19 elmapi 8843 . . . . . . . . . . . . . . . . 17 (𝑒 ∈ (β„•0 ↑m (1...π‘˜)) β†’ 𝑒:(1...π‘˜)βŸΆβ„•0)
20 fzss2 13541 . . . . . . . . . . . . . . . . 17 (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ (1...𝑁) βŠ† (1...π‘˜))
21 fssres 6758 . . . . . . . . . . . . . . . . 17 ((𝑒:(1...π‘˜)βŸΆβ„•0 ∧ (1...𝑁) βŠ† (1...π‘˜)) β†’ (𝑒 β†Ύ (1...𝑁)):(1...𝑁)βŸΆβ„•0)
2219, 20, 21syl2anr 598 . . . . . . . . . . . . . . . 16 ((π‘˜ ∈ (β„€β‰₯β€˜π‘) ∧ 𝑒 ∈ (β„•0 ↑m (1...π‘˜))) β†’ (𝑒 β†Ύ (1...𝑁)):(1...𝑁)βŸΆβ„•0)
23 nn0ex 12478 . . . . . . . . . . . . . . . . 17 β„•0 ∈ V
24 ovex 7442 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
2523, 24elmap 8865 . . . . . . . . . . . . . . . 16 ((𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)) ↔ (𝑒 β†Ύ (1...𝑁)):(1...𝑁)βŸΆβ„•0)
2622, 25sylibr 233 . . . . . . . . . . . . . . 15 ((π‘˜ ∈ (β„€β‰₯β€˜π‘) ∧ 𝑒 ∈ (β„•0 ↑m (1...π‘˜))) β†’ (𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)))
27 eleq1 2822 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑒 β†Ύ (1...𝑁)) β†’ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ↔ (𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁))))
2827adantr 482 . . . . . . . . . . . . . . 15 ((𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) β†’ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ↔ (𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁))))
2926, 28syl5ibrcom 246 . . . . . . . . . . . . . 14 ((π‘˜ ∈ (β„€β‰₯β€˜π‘) ∧ 𝑒 ∈ (β„•0 ↑m (1...π‘˜))) β†’ ((𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) β†’ 𝑑 ∈ (β„•0 ↑m (1...𝑁))))
3029rexlimdva 3156 . . . . . . . . . . . . 13 (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ (βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) β†’ 𝑑 ∈ (β„•0 ↑m (1...𝑁))))
3130abssdv 4066 . . . . . . . . . . . 12 (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} βŠ† (β„•0 ↑m (1...𝑁)))
3215elpw2 5346 . . . . . . . . . . . 12 ({𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ∈ 𝒫 (β„•0 ↑m (1...𝑁)) ↔ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} βŠ† (β„•0 ↑m (1...𝑁)))
3331, 32sylibr 233 . . . . . . . . . . 11 (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ∈ 𝒫 (β„•0 ↑m (1...𝑁)))
34 eleq1 2822 . . . . . . . . . . 11 (𝑑 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} β†’ (𝑑 ∈ 𝒫 (β„•0 ↑m (1...𝑁)) ↔ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ∈ 𝒫 (β„•0 ↑m (1...𝑁))))
3533, 34syl5ibrcom 246 . . . . . . . . . 10 (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ (𝑑 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} β†’ 𝑑 ∈ 𝒫 (β„•0 ↑m (1...𝑁))))
3635rexlimdvw 3161 . . . . . . . . 9 (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ (βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜))𝑑 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} β†’ 𝑑 ∈ 𝒫 (β„•0 ↑m (1...𝑁))))
3736rexlimiv 3149 . . . . . . . 8 (βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜))𝑑 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} β†’ 𝑑 ∈ 𝒫 (β„•0 ↑m (1...𝑁)))
3837abssi 4068 . . . . . . 7 {𝑑 ∣ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜))𝑑 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}} βŠ† 𝒫 (β„•0 ↑m (1...𝑁))
3918, 38eqsstri 4017 . . . . . 6 ran (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}) βŠ† 𝒫 (β„•0 ↑m (1...𝑁))
4016, 39ssexi 5323 . . . . 5 ran (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}) ∈ V
4114, 1, 40fvmpt 6999 . . . 4 (𝑁 ∈ β„•0 β†’ (Diophβ€˜π‘) = ran (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
4241eleq2d 2820 . . 3 (𝑁 ∈ β„•0 β†’ (𝐷 ∈ (Diophβ€˜π‘) ↔ 𝐷 ∈ ran (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})))
43 ovex 7442 . . . . . 6 (β„•0 ↑m (1...π‘˜)) ∈ V
4443abrexex 7949 . . . . 5 {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))𝑑 = (𝑒 β†Ύ (1...𝑁))} ∈ V
45 simpl 484 . . . . . . 7 ((𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) β†’ 𝑑 = (𝑒 β†Ύ (1...𝑁)))
4645reximi 3085 . . . . . 6 (βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) β†’ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))𝑑 = (𝑒 β†Ύ (1...𝑁)))
4746ss2abi 4064 . . . . 5 {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} βŠ† {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))𝑑 = (𝑒 β†Ύ (1...𝑁))}
4844, 47ssexi 5323 . . . 4 {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ∈ V
4917, 48elrnmpo 7545 . . 3 (𝐷 ∈ ran (π‘˜ ∈ (β„€β‰₯β€˜π‘), 𝑝 ∈ (mzPolyβ€˜(1...π‘˜)) ↦ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}) ↔ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜))𝐷 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
5042, 49bitrdi 287 . 2 (𝑁 ∈ β„•0 β†’ (𝐷 ∈ (Diophβ€˜π‘) ↔ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜))𝐷 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
514, 50biadanii 821 1 (𝐷 ∈ (Diophβ€˜π‘) ↔ (𝑁 ∈ β„•0 ∧ βˆƒπ‘˜ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘˜))𝐷 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (1...π‘˜))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071   βŠ† wss 3949  π’« cpw 4603  dom cdm 5677  ran crn 5678   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411   ↑m cmap 8820  0cc0 11110  1c1 11111  β„•0cn0 12472  β„€β‰₯cuz 12822  ...cfz 13484  mzPolycmzp 41460  Diophcdioph 41493
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-addcl 11170  ax-pre-lttri 11184  ax-pre-lttrn 11185
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-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-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-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-dioph 41494
This theorem is referenced by:  eldioph  41496  eldioph2b  41501  eldiophelnn0  41502
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