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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orngmullt | Structured version Visualization version GIF version |
Description: In an ordered ring, the strict ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
Ref | Expression |
---|---|
orngmullt.b | β’ π΅ = (Baseβπ ) |
orngmullt.t | β’ Β· = (.rβπ ) |
orngmullt.0 | β’ 0 = (0gβπ ) |
orngmullt.l | β’ < = (ltβπ ) |
orngmullt.1 | β’ (π β π β oRing) |
orngmullt.4 | β’ (π β π β DivRing) |
orngmullt.2 | β’ (π β π β π΅) |
orngmullt.3 | β’ (π β π β π΅) |
orngmullt.x | β’ (π β 0 < π) |
orngmullt.y | β’ (π β 0 < π) |
Ref | Expression |
---|---|
orngmullt | β’ (π β 0 < (π Β· π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orngmullt.1 | . . 3 β’ (π β π β oRing) | |
2 | orngmullt.2 | . . 3 β’ (π β π β π΅) | |
3 | orngmullt.x | . . . . 5 β’ (π β 0 < π) | |
4 | orngring 33063 | . . . . . . 7 β’ (π β oRing β π β Ring) | |
5 | ringgrp 20182 | . . . . . . 7 β’ (π β Ring β π β Grp) | |
6 | orngmullt.b | . . . . . . . 8 β’ π΅ = (Baseβπ ) | |
7 | orngmullt.0 | . . . . . . . 8 β’ 0 = (0gβπ ) | |
8 | 6, 7 | grpidcl 18926 | . . . . . . 7 β’ (π β Grp β 0 β π΅) |
9 | 1, 4, 5, 8 | 4syl 19 | . . . . . 6 β’ (π β 0 β π΅) |
10 | eqid 2725 | . . . . . . 7 β’ (leβπ ) = (leβπ ) | |
11 | orngmullt.l | . . . . . . 7 β’ < = (ltβπ ) | |
12 | 10, 11 | pltval 18323 | . . . . . 6 β’ ((π β oRing β§ 0 β π΅ β§ π β π΅) β ( 0 < π β ( 0 (leβπ )π β§ 0 β π))) |
13 | 1, 9, 2, 12 | syl3anc 1368 | . . . . 5 β’ (π β ( 0 < π β ( 0 (leβπ )π β§ 0 β π))) |
14 | 3, 13 | mpbid 231 | . . . 4 β’ (π β ( 0 (leβπ )π β§ 0 β π)) |
15 | 14 | simpld 493 | . . 3 β’ (π β 0 (leβπ )π) |
16 | orngmullt.3 | . . 3 β’ (π β π β π΅) | |
17 | orngmullt.y | . . . . 5 β’ (π β 0 < π) | |
18 | 10, 11 | pltval 18323 | . . . . . 6 β’ ((π β oRing β§ 0 β π΅ β§ π β π΅) β ( 0 < π β ( 0 (leβπ )π β§ 0 β π))) |
19 | 1, 9, 16, 18 | syl3anc 1368 | . . . . 5 β’ (π β ( 0 < π β ( 0 (leβπ )π β§ 0 β π))) |
20 | 17, 19 | mpbid 231 | . . . 4 β’ (π β ( 0 (leβπ )π β§ 0 β π)) |
21 | 20 | simpld 493 | . . 3 β’ (π β 0 (leβπ )π) |
22 | orngmullt.t | . . . 4 β’ Β· = (.rβπ ) | |
23 | 6, 10, 7, 22 | orngmul 33066 | . . 3 β’ ((π β oRing β§ (π β π΅ β§ 0 (leβπ )π) β§ (π β π΅ β§ 0 (leβπ )π)) β 0 (leβπ )(π Β· π)) |
24 | 1, 2, 15, 16, 21, 23 | syl122anc 1376 | . 2 β’ (π β 0 (leβπ )(π Β· π)) |
25 | 14 | simprd 494 | . . . . 5 β’ (π β 0 β π) |
26 | 25 | necomd 2986 | . . . 4 β’ (π β π β 0 ) |
27 | 20 | simprd 494 | . . . . 5 β’ (π β 0 β π) |
28 | 27 | necomd 2986 | . . . 4 β’ (π β π β 0 ) |
29 | orngmullt.4 | . . . . 5 β’ (π β π β DivRing) | |
30 | 6, 7, 22, 29, 2, 16 | drngmulne0 20658 | . . . 4 β’ (π β ((π Β· π) β 0 β (π β 0 β§ π β 0 ))) |
31 | 26, 28, 30 | mpbir2and 711 | . . 3 β’ (π β (π Β· π) β 0 ) |
32 | 31 | necomd 2986 | . 2 β’ (π β 0 β (π Β· π)) |
33 | 1, 4 | syl 17 | . . . 4 β’ (π β π β Ring) |
34 | 6, 22 | ringcl 20194 | . . . 4 β’ ((π β Ring β§ π β π΅ β§ π β π΅) β (π Β· π) β π΅) |
35 | 33, 2, 16, 34 | syl3anc 1368 | . . 3 β’ (π β (π Β· π) β π΅) |
36 | 10, 11 | pltval 18323 | . . 3 β’ ((π β oRing β§ 0 β π΅ β§ (π Β· π) β π΅) β ( 0 < (π Β· π) β ( 0 (leβπ )(π Β· π) β§ 0 β (π Β· π)))) |
37 | 1, 9, 35, 36 | syl3anc 1368 | . 2 β’ (π β ( 0 < (π Β· π) β ( 0 (leβπ )(π Β· π) β§ 0 β (π Β· π)))) |
38 | 24, 32, 37 | mpbir2and 711 | 1 β’ (π β 0 < (π Β· π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 class class class wbr 5143 βcfv 6543 (class class class)co 7416 Basecbs 17179 .rcmulr 17233 lecple 17239 0gc0g 17420 ltcplt 18299 Grpcgrp 18894 Ringcrg 20177 DivRingcdr 20628 oRingcorng 33058 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-0g 17422 df-plt 18321 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-drng 20630 df-orng 33060 |
This theorem is referenced by: (None) |
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