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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldioph4b Structured version   Visualization version   GIF version
Theorem eldioph4b 42296
Description: Membership in Dioph expressed using a quantified union to add witness variables instead of a restriction to remove them. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Hypotheses
Ref Expression
eldioph4b.a π‘Š ∈ V
eldioph4b.b Β¬ π‘Š ∈ Fin
eldioph4b.c (π‘Š ∩ β„•) = βˆ…
Assertion
Ref Expression
eldioph4b (𝑆 ∈ (Diophβ€˜π‘) ↔ (𝑁 ∈ β„•0 ∧ βˆƒπ‘ ∈ (mzPolyβ€˜(π‘Š βˆͺ (1...𝑁)))𝑆 = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0}))
Distinct variable groups:   π‘Š,𝑝,𝑑,𝑀   𝑆,𝑝,𝑑,𝑀   𝑁,𝑝,𝑑,𝑀
Proof of Theorem eldioph4b
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 eldiophelnn0 42249 . 2 (𝑆 ∈ (Diophβ€˜π‘) β†’ 𝑁 ∈ β„•0)
2 eldioph4b.a . . . . . 6 π‘Š ∈ V
3 ovex 7449 . . . . . 6 (1...𝑁) ∈ V
42, 3unex 7746 . . . . 5 (π‘Š βˆͺ (1...𝑁)) ∈ V
54jctr 523 . . . 4 (𝑁 ∈ β„•0 β†’ (𝑁 ∈ β„•0 ∧ (π‘Š βˆͺ (1...𝑁)) ∈ V))
6 eldioph4b.b . . . . . . 7 Β¬ π‘Š ∈ Fin
76intnanr 486 . . . . . 6 Β¬ (π‘Š ∈ Fin ∧ (1...𝑁) ∈ Fin)
8 unfir 9337 . . . . . 6 ((π‘Š βˆͺ (1...𝑁)) ∈ Fin β†’ (π‘Š ∈ Fin ∧ (1...𝑁) ∈ Fin))
97, 8mto 196 . . . . 5 Β¬ (π‘Š βˆͺ (1...𝑁)) ∈ Fin
10 ssun2 4167 . . . . 5 (1...𝑁) βŠ† (π‘Š βˆͺ (1...𝑁))
119, 10pm3.2i 469 . . . 4 (Β¬ (π‘Š βˆͺ (1...𝑁)) ∈ Fin ∧ (1...𝑁) βŠ† (π‘Š βˆͺ (1...𝑁)))
12 eldioph2b 42248 . . . 4 (((𝑁 ∈ β„•0 ∧ (π‘Š βˆͺ (1...𝑁)) ∈ V) ∧ (Β¬ (π‘Š βˆͺ (1...𝑁)) ∈ Fin ∧ (1...𝑁) βŠ† (π‘Š βˆͺ (1...𝑁)))) β†’ (𝑆 ∈ (Diophβ€˜π‘) ↔ βˆƒπ‘ ∈ (mzPolyβ€˜(π‘Š βˆͺ (1...𝑁)))𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
135, 11, 12sylancl 584 . . 3 (𝑁 ∈ β„•0 β†’ (𝑆 ∈ (Diophβ€˜π‘) ↔ βˆƒπ‘ ∈ (mzPolyβ€˜(π‘Š βˆͺ (1...𝑁)))𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
14 elmapssres 8884 . . . . . . . . . . . . . . 15 ((𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) ∧ (1...𝑁) βŠ† (π‘Š βˆͺ (1...𝑁))) β†’ (𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)))
1510, 14mpan2 689 . . . . . . . . . . . . . 14 (𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) β†’ (𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)))
1615adantr 479 . . . . . . . . . . . . 13 ((𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) ∧ (π‘β€˜π‘’) = 0) β†’ (𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)))
17 ssun1 4166 . . . . . . . . . . . . . . . 16 π‘Š βŠ† (π‘Š βˆͺ (1...𝑁))
18 elmapssres 8884 . . . . . . . . . . . . . . . 16 ((𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) ∧ π‘Š βŠ† (π‘Š βˆͺ (1...𝑁))) β†’ (𝑒 β†Ύ π‘Š) ∈ (β„•0 ↑m π‘Š))
1917, 18mpan2 689 . . . . . . . . . . . . . . 15 (𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) β†’ (𝑒 β†Ύ π‘Š) ∈ (β„•0 ↑m π‘Š))
2019adantr 479 . . . . . . . . . . . . . 14 ((𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) ∧ (π‘β€˜π‘’) = 0) β†’ (𝑒 β†Ύ π‘Š) ∈ (β„•0 ↑m π‘Š))
21 uncom 4146 . . . . . . . . . . . . . . . . . 18 ((𝑒 β†Ύ (1...𝑁)) βˆͺ (𝑒 β†Ύ π‘Š)) = ((𝑒 β†Ύ π‘Š) βˆͺ (𝑒 β†Ύ (1...𝑁)))
22 resundi 5993 . . . . . . . . . . . . . . . . . 18 (𝑒 β†Ύ (π‘Š βˆͺ (1...𝑁))) = ((𝑒 β†Ύ π‘Š) βˆͺ (𝑒 β†Ύ (1...𝑁)))
2321, 22eqtr4i 2756 . . . . . . . . . . . . . . . . 17 ((𝑒 β†Ύ (1...𝑁)) βˆͺ (𝑒 β†Ύ π‘Š)) = (𝑒 β†Ύ (π‘Š βˆͺ (1...𝑁)))
24 elmapi 8866 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) β†’ 𝑒:(π‘Š βˆͺ (1...𝑁))βŸΆβ„•0)
25 ffn 6717 . . . . . . . . . . . . . . . . . 18 (𝑒:(π‘Š βˆͺ (1...𝑁))βŸΆβ„•0 β†’ 𝑒 Fn (π‘Š βˆͺ (1...𝑁)))
26 fnresdm 6669 . . . . . . . . . . . . . . . . . 18 (𝑒 Fn (π‘Š βˆͺ (1...𝑁)) β†’ (𝑒 β†Ύ (π‘Š βˆͺ (1...𝑁))) = 𝑒)
2724, 25, 263syl 18 . . . . . . . . . . . . . . . . 17 (𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) β†’ (𝑒 β†Ύ (π‘Š βˆͺ (1...𝑁))) = 𝑒)
2823, 27eqtrid 2777 . . . . . . . . . . . . . . . 16 (𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) β†’ ((𝑒 β†Ύ (1...𝑁)) βˆͺ (𝑒 β†Ύ π‘Š)) = 𝑒)
2928fveqeq2d 6900 . . . . . . . . . . . . . . 15 (𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) β†’ ((π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ (𝑒 β†Ύ π‘Š))) = 0 ↔ (π‘β€˜π‘’) = 0))
3029biimpar 476 . . . . . . . . . . . . . 14 ((𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) ∧ (π‘β€˜π‘’) = 0) β†’ (π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ (𝑒 β†Ύ π‘Š))) = 0)
31 uneq2 4150 . . . . . . . . . . . . . . . 16 (𝑀 = (𝑒 β†Ύ π‘Š) β†’ ((𝑒 β†Ύ (1...𝑁)) βˆͺ 𝑀) = ((𝑒 β†Ύ (1...𝑁)) βˆͺ (𝑒 β†Ύ π‘Š)))
3231fveqeq2d 6900 . . . . . . . . . . . . . . 15 (𝑀 = (𝑒 β†Ύ π‘Š) β†’ ((π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ 𝑀)) = 0 ↔ (π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ (𝑒 β†Ύ π‘Š))) = 0))
3332rspcev 3601 . . . . . . . . . . . . . 14 (((𝑒 β†Ύ π‘Š) ∈ (β„•0 ↑m π‘Š) ∧ (π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ (𝑒 β†Ύ π‘Š))) = 0) β†’ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ 𝑀)) = 0)
3420, 30, 33syl2anc 582 . . . . . . . . . . . . 13 ((𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) ∧ (π‘β€˜π‘’) = 0) β†’ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ 𝑀)) = 0)
3516, 34jca 510 . . . . . . . . . . . 12 ((𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) ∧ (π‘β€˜π‘’) = 0) β†’ ((𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ 𝑀)) = 0))
36 eleq1 2813 . . . . . . . . . . . . 13 (𝑑 = (𝑒 β†Ύ (1...𝑁)) β†’ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ↔ (𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁))))
37 uneq1 4149 . . . . . . . . . . . . . . 15 (𝑑 = (𝑒 β†Ύ (1...𝑁)) β†’ (𝑑 βˆͺ 𝑀) = ((𝑒 β†Ύ (1...𝑁)) βˆͺ 𝑀))
3837fveqeq2d 6900 . . . . . . . . . . . . . 14 (𝑑 = (𝑒 β†Ύ (1...𝑁)) β†’ ((π‘β€˜(𝑑 βˆͺ 𝑀)) = 0 ↔ (π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ 𝑀)) = 0))
3938rexbidv 3169 . . . . . . . . . . . . 13 (𝑑 = (𝑒 β†Ύ (1...𝑁)) β†’ (βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0 ↔ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ 𝑀)) = 0))
4036, 39anbi12d 630 . . . . . . . . . . . 12 (𝑑 = (𝑒 β†Ύ (1...𝑁)) β†’ ((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0) ↔ ((𝑒 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜((𝑒 β†Ύ (1...𝑁)) βˆͺ 𝑀)) = 0)))
4135, 40syl5ibrcom 246 . . . . . . . . . . 11 ((𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) ∧ (π‘β€˜π‘’) = 0) β†’ (𝑑 = (𝑒 β†Ύ (1...𝑁)) β†’ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0)))
4241expimpd 452 . . . . . . . . . 10 (𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) β†’ (((π‘β€˜π‘’) = 0 ∧ 𝑑 = (𝑒 β†Ύ (1...𝑁))) β†’ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0)))
4342ancomsd 464 . . . . . . . . 9 (𝑒 ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) β†’ ((𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) β†’ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0)))
4443rexlimiv 3138 . . . . . . . 8 (βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) β†’ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0))
45 uncom 4146 . . . . . . . . . . . 12 (𝑑 βˆͺ 𝑀) = (𝑀 βˆͺ 𝑑)
46 fz1ssnn 13564 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) βŠ† β„•
47 sslin 4229 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑁) βŠ† β„• β†’ (π‘Š ∩ (1...𝑁)) βŠ† (π‘Š ∩ β„•))
4846, 47ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (π‘Š ∩ (1...𝑁)) βŠ† (π‘Š ∩ β„•)
49 eldioph4b.c . . . . . . . . . . . . . . . . . . 19 (π‘Š ∩ β„•) = βˆ…
5048, 49sseqtri 4009 . . . . . . . . . . . . . . . . . 18 (π‘Š ∩ (1...𝑁)) βŠ† βˆ…
51 ss0 4394 . . . . . . . . . . . . . . . . . 18 ((π‘Š ∩ (1...𝑁)) βŠ† βˆ… β†’ (π‘Š ∩ (1...𝑁)) = βˆ…)
5250, 51ax-mp 5 . . . . . . . . . . . . . . . . 17 (π‘Š ∩ (1...𝑁)) = βˆ…
5352reseq2i 5976 . . . . . . . . . . . . . . . 16 (𝑀 β†Ύ (π‘Š ∩ (1...𝑁))) = (𝑀 β†Ύ βˆ…)
54 res0 5983 . . . . . . . . . . . . . . . 16 (𝑀 β†Ύ βˆ…) = βˆ…
5553, 54eqtri 2753 . . . . . . . . . . . . . . 15 (𝑀 β†Ύ (π‘Š ∩ (1...𝑁))) = βˆ…
5652reseq2i 5976 . . . . . . . . . . . . . . . 16 (𝑑 β†Ύ (π‘Š ∩ (1...𝑁))) = (𝑑 β†Ύ βˆ…)
57 res0 5983 . . . . . . . . . . . . . . . 16 (𝑑 β†Ύ βˆ…) = βˆ…
5856, 57eqtri 2753 . . . . . . . . . . . . . . 15 (𝑑 β†Ύ (π‘Š ∩ (1...𝑁))) = βˆ…
5955, 58eqtr4i 2756 . . . . . . . . . . . . . 14 (𝑀 β†Ύ (π‘Š ∩ (1...𝑁))) = (𝑑 β†Ύ (π‘Š ∩ (1...𝑁)))
60 elmapresaun 8897 . . . . . . . . . . . . . 14 ((𝑀 ∈ (β„•0 ↑m π‘Š) ∧ 𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ (𝑀 β†Ύ (π‘Š ∩ (1...𝑁))) = (𝑑 β†Ύ (π‘Š ∩ (1...𝑁)))) β†’ (𝑀 βˆͺ 𝑑) ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))))
6159, 60mp3an3 1446 . . . . . . . . . . . . 13 ((𝑀 ∈ (β„•0 ↑m π‘Š) ∧ 𝑑 ∈ (β„•0 ↑m (1...𝑁))) β†’ (𝑀 βˆͺ 𝑑) ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))))
6261ancoms 457 . . . . . . . . . . . 12 ((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑀 ∈ (β„•0 ↑m π‘Š)) β†’ (𝑀 βˆͺ 𝑑) ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))))
6345, 62eqeltrid 2829 . . . . . . . . . . 11 ((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑀 ∈ (β„•0 ↑m π‘Š)) β†’ (𝑑 βˆͺ 𝑀) ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))))
6463adantr 479 . . . . . . . . . 10 (((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑀 ∈ (β„•0 ↑m π‘Š)) ∧ (π‘β€˜(𝑑 βˆͺ 𝑀)) = 0) β†’ (𝑑 βˆͺ 𝑀) ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))))
6545reseq1i 5975 . . . . . . . . . . . 12 ((𝑑 βˆͺ 𝑀) β†Ύ (1...𝑁)) = ((𝑀 βˆͺ 𝑑) β†Ύ (1...𝑁))
66 elmapresaunres2 42256 . . . . . . . . . . . . . 14 ((𝑀 ∈ (β„•0 ↑m π‘Š) ∧ 𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ (𝑀 β†Ύ (π‘Š ∩ (1...𝑁))) = (𝑑 β†Ύ (π‘Š ∩ (1...𝑁)))) β†’ ((𝑀 βˆͺ 𝑑) β†Ύ (1...𝑁)) = 𝑑)
6759, 66mp3an3 1446 . . . . . . . . . . . . 13 ((𝑀 ∈ (β„•0 ↑m π‘Š) ∧ 𝑑 ∈ (β„•0 ↑m (1...𝑁))) β†’ ((𝑀 βˆͺ 𝑑) β†Ύ (1...𝑁)) = 𝑑)
6867ancoms 457 . . . . . . . . . . . 12 ((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑀 ∈ (β„•0 ↑m π‘Š)) β†’ ((𝑀 βˆͺ 𝑑) β†Ύ (1...𝑁)) = 𝑑)
6965, 68eqtr2id 2778 . . . . . . . . . . 11 ((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑀 ∈ (β„•0 ↑m π‘Š)) β†’ 𝑑 = ((𝑑 βˆͺ 𝑀) β†Ύ (1...𝑁)))
7069adantr 479 . . . . . . . . . 10 (((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑀 ∈ (β„•0 ↑m π‘Š)) ∧ (π‘β€˜(𝑑 βˆͺ 𝑀)) = 0) β†’ 𝑑 = ((𝑑 βˆͺ 𝑀) β†Ύ (1...𝑁)))
71 simpr 483 . . . . . . . . . 10 (((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑀 ∈ (β„•0 ↑m π‘Š)) ∧ (π‘β€˜(𝑑 βˆͺ 𝑀)) = 0) β†’ (π‘β€˜(𝑑 βˆͺ 𝑀)) = 0)
72 reseq1 5973 . . . . . . . . . . . . 13 (𝑒 = (𝑑 βˆͺ 𝑀) β†’ (𝑒 β†Ύ (1...𝑁)) = ((𝑑 βˆͺ 𝑀) β†Ύ (1...𝑁)))
7372eqeq2d 2736 . . . . . . . . . . . 12 (𝑒 = (𝑑 βˆͺ 𝑀) β†’ (𝑑 = (𝑒 β†Ύ (1...𝑁)) ↔ 𝑑 = ((𝑑 βˆͺ 𝑀) β†Ύ (1...𝑁))))
74 fveqeq2 6901 . . . . . . . . . . . 12 (𝑒 = (𝑑 βˆͺ 𝑀) β†’ ((π‘β€˜π‘’) = 0 ↔ (π‘β€˜(𝑑 βˆͺ 𝑀)) = 0))
7573, 74anbi12d 630 . . . . . . . . . . 11 (𝑒 = (𝑑 βˆͺ 𝑀) β†’ ((𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) ↔ (𝑑 = ((𝑑 βˆͺ 𝑀) β†Ύ (1...𝑁)) ∧ (π‘β€˜(𝑑 βˆͺ 𝑀)) = 0)))
7675rspcev 3601 . . . . . . . . . 10 (((𝑑 βˆͺ 𝑀) ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁))) ∧ (𝑑 = ((𝑑 βˆͺ 𝑀) β†Ύ (1...𝑁)) ∧ (π‘β€˜(𝑑 βˆͺ 𝑀)) = 0)) β†’ βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0))
7764, 70, 71, 76syl12anc 835 . . . . . . . . 9 (((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑀 ∈ (β„•0 ↑m π‘Š)) ∧ (π‘β€˜(𝑑 βˆͺ 𝑀)) = 0) β†’ βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0))
7877r19.29an 3148 . . . . . . . 8 ((𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0) β†’ βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0))
7944, 78impbii 208 . . . . . . 7 (βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) ↔ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0))
8079abbii 2795 . . . . . 6 {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} = {𝑑 ∣ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0)}
81 df-rab 3420 . . . . . 6 {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0} = {𝑑 ∣ (𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0)}
8280, 81eqtr4i 2756 . . . . 5 {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0}
8382eqeq2i 2738 . . . 4 (𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ↔ 𝑆 = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0})
8483rexbii 3084 . . 3 (βˆƒπ‘ ∈ (mzPolyβ€˜(π‘Š βˆͺ (1...𝑁)))𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m (π‘Š βˆͺ (1...𝑁)))(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ↔ βˆƒπ‘ ∈ (mzPolyβ€˜(π‘Š βˆͺ (1...𝑁)))𝑆 = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0})
8513, 84bitrdi 286 . 2 (𝑁 ∈ β„•0 β†’ (𝑆 ∈ (Diophβ€˜π‘) ↔ βˆƒπ‘ ∈ (mzPolyβ€˜(π‘Š βˆͺ (1...𝑁)))𝑆 = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0}))
861, 85biadanii 820 1 (𝑆 ∈ (Diophβ€˜π‘) ↔ (𝑁 ∈ β„•0 ∧ βˆƒπ‘ ∈ (mzPolyβ€˜(π‘Š βˆͺ (1...𝑁)))𝑆 = {𝑑 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘€ ∈ (β„•0 ↑m π‘Š)(π‘β€˜(𝑑 βˆͺ 𝑀)) = 0}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆƒwrex 3060  {crab 3419  Vcvv 3463   βˆͺ cun 3937   ∩ cin 3938   βŠ† wss 3939  βˆ…c0 4318   β†Ύ cres 5674   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7416   ↑m cmap 8843  Fincfn 8962  0cc0 11138  1c1 11139  β„•cn 12242  β„•0cn0 12502  ...cfz 13516  mzPolycmzp 42207  Diophcdioph 42240
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 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
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-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  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-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 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-hash 14322  df-mzpcl 42208  df-mzp 42209  df-dioph 42241
This theorem is referenced by:  eldioph4i  42297  diophren  42298
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