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Mirrors > Home > MPE Home > Th. List > qnumval | Structured version Visualization version GIF version |
Description: Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
qnumval | β’ (π΄ β β β (numerβπ΄) = (1st β(β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π΄ = ((1st βπ₯) / (2nd βπ₯)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2732 | . . . . 5 β’ (π = π΄ β (π = ((1st βπ₯) / (2nd βπ₯)) β π΄ = ((1st βπ₯) / (2nd βπ₯)))) | |
2 | 1 | anbi2d 628 | . . . 4 β’ (π = π΄ β ((((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π = ((1st βπ₯) / (2nd βπ₯))) β (((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π΄ = ((1st βπ₯) / (2nd βπ₯))))) |
3 | 2 | riotabidv 7384 | . . 3 β’ (π = π΄ β (β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π = ((1st βπ₯) / (2nd βπ₯)))) = (β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π΄ = ((1st βπ₯) / (2nd βπ₯))))) |
4 | 3 | fveq2d 6906 | . 2 β’ (π = π΄ β (1st β(β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π = ((1st βπ₯) / (2nd βπ₯))))) = (1st β(β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π΄ = ((1st βπ₯) / (2nd βπ₯)))))) |
5 | df-numer 16714 | . 2 β’ numer = (π β β β¦ (1st β(β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π = ((1st βπ₯) / (2nd βπ₯)))))) | |
6 | fvex 6915 | . 2 β’ (1st β(β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π΄ = ((1st βπ₯) / (2nd βπ₯))))) β V | |
7 | 4, 5, 6 | fvmpt 7010 | 1 β’ (π΄ β β β (numerβπ΄) = (1st β(β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π΄ = ((1st βπ₯) / (2nd βπ₯)))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Γ cxp 5680 βcfv 6553 β©crio 7381 (class class class)co 7426 1st c1st 7997 2nd c2nd 7998 1c1 11147 / cdiv 11909 βcn 12250 β€cz 12596 βcq 12970 gcd cgcd 16476 numercnumer 16712 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-riota 7382 df-numer 16714 |
This theorem is referenced by: qnumdencl 16718 fnum 16721 qnumdenbi 16723 |
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