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 31190 | . . . . . . 7 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring) | |
5 | ringgrp 19539 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
6 | orngmullt.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
7 | orngmullt.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
8 | 6, 7 | grpidcl 18367 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
9 | 1, 4, 5, 8 | 4syl 19 | . . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐵) |
10 | eqid 2734 | . . . . . . 7 ⊢ (le‘𝑅) = (le‘𝑅) | |
11 | orngmullt.l | . . . . . . 7 ⊢ < = (lt‘𝑅) | |
12 | 10, 11 | pltval 17810 | . . . . . 6 ⊢ ((𝑅 ∈ oRing ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝑅)𝑋 ∧ 0 ≠ 𝑋))) |
13 | 1, 9, 2, 12 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → ( 0 < 𝑋 ↔ ( 0 (le‘𝑅)𝑋 ∧ 0 ≠ 𝑋))) |
14 | 3, 13 | mpbid 235 | . . . 4 ⊢ (𝜑 → ( 0 (le‘𝑅)𝑋 ∧ 0 ≠ 𝑋)) |
15 | 14 | simpld 498 | . . 3 ⊢ (𝜑 → 0 (le‘𝑅)𝑋) |
16 | orngmullt.3 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
17 | orngmullt.y | . . . . 5 ⊢ (𝜑 → 0 < 𝑌) | |
18 | 10, 11 | pltval 17810 | . . . . . 6 ⊢ ((𝑅 ∈ oRing ∧ 0 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 < 𝑌 ↔ ( 0 (le‘𝑅)𝑌 ∧ 0 ≠ 𝑌))) |
19 | 1, 9, 16, 18 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → ( 0 < 𝑌 ↔ ( 0 (le‘𝑅)𝑌 ∧ 0 ≠ 𝑌))) |
20 | 17, 19 | mpbid 235 | . . . 4 ⊢ (𝜑 → ( 0 (le‘𝑅)𝑌 ∧ 0 ≠ 𝑌)) |
21 | 20 | simpld 498 | . . 3 ⊢ (𝜑 → 0 (le‘𝑅)𝑌) |
22 | orngmullt.t | . . . 4 ⊢ · = (.r‘𝑅) | |
23 | 6, 10, 7, 22 | orngmul 31193 | . . 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 1381 | . 2 ⊢ (𝜑 → 0 (le‘𝑅)(𝑋 · 𝑌)) |
25 | 14 | simprd 499 | . . . . 5 ⊢ (𝜑 → 0 ≠ 𝑋) |
26 | 25 | necomd 2990 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
27 | 20 | simprd 499 | . . . . 5 ⊢ (𝜑 → 0 ≠ 𝑌) |
28 | 27 | necomd 2990 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 0 ) |
29 | orngmullt.4 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
30 | 6, 7, 22, 29, 2, 16 | drngmulne0 19761 | . . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))) |
31 | 26, 28, 30 | mpbir2and 713 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ≠ 0 ) |
32 | 31 | necomd 2990 | . 2 ⊢ (𝜑 → 0 ≠ (𝑋 · 𝑌)) |
33 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
34 | 6, 22 | ringcl 19551 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
35 | 33, 2, 16, 34 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
36 | 10, 11 | pltval 17810 | . . 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 1373 | . 2 ⊢ (𝜑 → ( 0 < (𝑋 · 𝑌) ↔ ( 0 (le‘𝑅)(𝑋 · 𝑌) ∧ 0 ≠ (𝑋 · 𝑌)))) |
38 | 24, 32, 37 | mpbir2and 713 | 1 ⊢ (𝜑 → 0 < (𝑋 · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 .rcmulr 16768 lecple 16774 0gc0g 16916 ltcplt 17787 Grpcgrp 18337 Ringcrg 19534 DivRingcdr 19739 oRingcorng 31185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-0g 16918 df-plt 17808 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-mgp 19477 df-ur 19489 df-ring 19536 df-oppr 19613 df-dvdsr 19631 df-unit 19632 df-invr 19662 df-drng 19741 df-orng 31187 |
This theorem is referenced by: (None) |
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