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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 2730 | . . . . 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 7362 | . . 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 6888 | . 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 16677 | . 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 6897 | . 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 6991 | 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 395 = wceq 1533 β wcel 2098 Γ cxp 5667 βcfv 6536 β©crio 7359 (class class class)co 7404 1st c1st 7969 2nd c2nd 7970 1c1 11110 / cdiv 11872 βcn 12213 β€cz 12559 βcq 12933 gcd cgcd 16439 numercnumer 16675 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-riota 7360 df-numer 16677 |
This theorem is referenced by: qnumdencl 16681 fnum 16684 qnumdenbi 16686 |
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