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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 | ⊢ 𝐵 = (Base‘𝑅) |
orngmullt.t | ⊢ · = (.r‘𝑅) |
orngmullt.0 | ⊢ 0 = (0g‘𝑅) |
orngmullt.l | ⊢ < = (lt‘𝑅) |
orngmullt.1 | ⊢ (𝜑 → 𝑅 ∈ oRing) |
orngmullt.4 | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
orngmullt.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
orngmullt.3 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
orngmullt.x | ⊢ (𝜑 → 0 < 𝑋) |
orngmullt.y | ⊢ (𝜑 → 0 < 𝑌) |
Ref | Expression |
---|---|
orngmullt | ⊢ (𝜑 → 0 < (𝑋 · 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orngmullt.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ oRing) | |
2 | orngmullt.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | orngmullt.x | . . . . 5 ⊢ (𝜑 → 0 < 𝑋) | |
4 | orngring 32095 | . . . . . . 7 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring) | |
5 | ringgrp 19969 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
6 | orngmullt.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
7 | orngmullt.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
8 | 6, 7 | grpidcl 18778 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
9 | 1, 4, 5, 8 | 4syl 19 | . . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐵) |
10 | eqid 2736 | . . . . . . 7 ⊢ (le‘𝑅) = (le‘𝑅) | |
11 | orngmullt.l | . . . . . . 7 ⊢ < = (lt‘𝑅) | |
12 | 10, 11 | pltval 18221 | . . . . . 6 ⊢ ((𝑅 ∈ oRing ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝑅)𝑋 ∧ 0 ≠ 𝑋))) |
13 | 1, 9, 2, 12 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → ( 0 < 𝑋 ↔ ( 0 (le‘𝑅)𝑋 ∧ 0 ≠ 𝑋))) |
14 | 3, 13 | mpbid 231 | . . . 4 ⊢ (𝜑 → ( 0 (le‘𝑅)𝑋 ∧ 0 ≠ 𝑋)) |
15 | 14 | simpld 495 | . . 3 ⊢ (𝜑 → 0 (le‘𝑅)𝑋) |
16 | orngmullt.3 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
17 | orngmullt.y | . . . . 5 ⊢ (𝜑 → 0 < 𝑌) | |
18 | 10, 11 | pltval 18221 | . . . . . 6 ⊢ ((𝑅 ∈ oRing ∧ 0 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 < 𝑌 ↔ ( 0 (le‘𝑅)𝑌 ∧ 0 ≠ 𝑌))) |
19 | 1, 9, 16, 18 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → ( 0 < 𝑌 ↔ ( 0 (le‘𝑅)𝑌 ∧ 0 ≠ 𝑌))) |
20 | 17, 19 | mpbid 231 | . . . 4 ⊢ (𝜑 → ( 0 (le‘𝑅)𝑌 ∧ 0 ≠ 𝑌)) |
21 | 20 | simpld 495 | . . 3 ⊢ (𝜑 → 0 (le‘𝑅)𝑌) |
22 | orngmullt.t | . . . 4 ⊢ · = (.r‘𝑅) | |
23 | 6, 10, 7, 22 | orngmul 32098 | . . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 (le‘𝑅)𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 (le‘𝑅)𝑌)) → 0 (le‘𝑅)(𝑋 · 𝑌)) |
24 | 1, 2, 15, 16, 21, 23 | syl122anc 1379 | . 2 ⊢ (𝜑 → 0 (le‘𝑅)(𝑋 · 𝑌)) |
25 | 14 | simprd 496 | . . . . 5 ⊢ (𝜑 → 0 ≠ 𝑋) |
26 | 25 | necomd 2999 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
27 | 20 | simprd 496 | . . . . 5 ⊢ (𝜑 → 0 ≠ 𝑌) |
28 | 27 | necomd 2999 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 0 ) |
29 | orngmullt.4 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
30 | 6, 7, 22, 29, 2, 16 | drngmulne0 20211 | . . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))) |
31 | 26, 28, 30 | mpbir2and 711 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ≠ 0 ) |
32 | 31 | necomd 2999 | . 2 ⊢ (𝜑 → 0 ≠ (𝑋 · 𝑌)) |
33 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
34 | 6, 22 | ringcl 19981 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
35 | 33, 2, 16, 34 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
36 | 10, 11 | pltval 18221 | . . 3 ⊢ ((𝑅 ∈ oRing ∧ 0 ∈ 𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐵) → ( 0 < (𝑋 · 𝑌) ↔ ( 0 (le‘𝑅)(𝑋 · 𝑌) ∧ 0 ≠ (𝑋 · 𝑌)))) |
37 | 1, 9, 35, 36 | syl3anc 1371 | . 2 ⊢ (𝜑 → ( 0 < (𝑋 · 𝑌) ↔ ( 0 (le‘𝑅)(𝑋 · 𝑌) ∧ 0 ≠ (𝑋 · 𝑌)))) |
38 | 24, 32, 37 | mpbir2and 711 | 1 ⊢ (𝜑 → 0 < (𝑋 · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 .rcmulr 17134 lecple 17140 0gc0g 17321 ltcplt 18197 Grpcgrp 18748 Ringcrg 19964 DivRingcdr 20185 oRingcorng 32090 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-0g 17323 df-plt 18219 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-drng 20187 df-orng 32092 |
This theorem is referenced by: (None) |
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