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Mirrors > Home > MPE Home > Th. List > qdenval | Structured version Visualization version GIF version |
Description: Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
qdenval | β’ (π΄ β β β (denomβπ΄) = (2nd 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 2737 | . . . . 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 7367 | . . 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 6896 | . 2 β’ (π = π΄ β (2nd β(β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π = ((1st βπ₯) / (2nd βπ₯))))) = (2nd β(β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π΄ = ((1st βπ₯) / (2nd βπ₯)))))) |
5 | df-denom 16672 | . 2 β’ denom = (π β β β¦ (2nd β(β©π₯ β (β€ Γ β)(((1st βπ₯) gcd (2nd βπ₯)) = 1 β§ π = ((1st βπ₯) / (2nd βπ₯)))))) | |
6 | fvex 6905 | . 2 β’ (2nd 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 6999 | 1 β’ (π΄ β β β (denomβπ΄) = (2nd 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 5675 βcfv 6544 β©crio 7364 (class class class)co 7409 1st c1st 7973 2nd c2nd 7974 1c1 11111 / cdiv 11871 βcn 12212 β€cz 12558 βcq 12932 gcd cgcd 16435 denomcdenom 16670 |
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-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-riota 7365 df-denom 16672 |
This theorem is referenced by: qnumdencl 16675 fden 16679 qnumdenbi 16680 |
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