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Mirrors > Home > MPE Home > Th. List > quotdgr | Structured version Visualization version GIF version |
Description: Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.) |
Ref | Expression |
---|---|
quotdgr.1 | β’ π = (πΉ βf β (πΊ βf Β· (πΉ quot πΊ))) |
Ref | Expression |
---|---|
quotdgr | β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β (π = 0π β¨ (degβπ ) < (degβπΊ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcl 11194 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯ + π¦) β β) | |
2 | 1 | adantl 482 | . . 3 β’ (((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β§ (π₯ β β β§ π¦ β β)) β (π₯ + π¦) β β) |
3 | mulcl 11196 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯ Β· π¦) β β) | |
4 | 3 | adantl 482 | . . 3 β’ (((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β§ (π₯ β β β§ π¦ β β)) β (π₯ Β· π¦) β β) |
5 | reccl 11883 | . . . 4 β’ ((π₯ β β β§ π₯ β 0) β (1 / π₯) β β) | |
6 | 5 | adantl 482 | . . 3 β’ (((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β§ (π₯ β β β§ π₯ β 0)) β (1 / π₯) β β) |
7 | neg1cn 12330 | . . . 4 β’ -1 β β | |
8 | 7 | a1i 11 | . . 3 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β -1 β β) |
9 | plyssc 25938 | . . . 4 β’ (Polyβπ) β (Polyββ) | |
10 | simp1 1136 | . . . 4 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β πΉ β (Polyβπ)) | |
11 | 9, 10 | sselid 3980 | . . 3 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β πΉ β (Polyββ)) |
12 | simp2 1137 | . . . 4 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β πΊ β (Polyβπ)) | |
13 | 9, 12 | sselid 3980 | . . 3 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β πΊ β (Polyββ)) |
14 | simp3 1138 | . . 3 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β πΊ β 0π) | |
15 | quotdgr.1 | . . 3 β’ π = (πΉ βf β (πΊ βf Β· (πΉ quot πΊ))) | |
16 | 2, 4, 6, 8, 11, 13, 14, 15 | quotlem 26037 | . 2 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β ((πΉ quot πΊ) β (Polyββ) β§ (π = 0π β¨ (degβπ ) < (degβπΊ)))) |
17 | 16 | simprd 496 | 1 β’ ((πΉ β (Polyβπ) β§ πΊ β (Polyβπ) β§ πΊ β 0π) β (π = 0π β¨ (degβπ ) < (degβπΊ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β¨ wo 845 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 class class class wbr 5148 βcfv 6543 (class class class)co 7411 βf cof 7670 βcc 11110 0cc0 11112 1c1 11113 + caddc 11115 Β· cmul 11117 < clt 11252 β cmin 11448 -cneg 11449 / cdiv 11875 0πc0p 25410 Polycply 25922 degcdgr 25925 quot cquot 26027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-0p 25411 df-ply 25926 df-coe 25928 df-dgr 25929 df-quot 26028 |
This theorem is referenced by: plyrem 26042 quotcan 26046 |
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