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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 2741 | . . . . 5 β’ (π = π΄ β (π = ((1st βπ₯) / (2nd βπ₯)) β π΄ = ((1st βπ₯) / (2nd βπ₯)))) | |
2 | 1 | anbi2d 630 | . . . 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 7320 | . . 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 6851 | . 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 16617 | . 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 6860 | . 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 6953 | 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 397 = wceq 1542 β wcel 2107 Γ cxp 5636 βcfv 6501 β©crio 7317 (class class class)co 7362 1st c1st 7924 2nd c2nd 7925 1c1 11059 / cdiv 11819 βcn 12160 β€cz 12506 βcq 12880 gcd cgcd 16381 numercnumer 16615 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-riota 7318 df-numer 16617 |
This theorem is referenced by: qnumdencl 16621 fnum 16624 qnumdenbi 16626 |
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