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Theorem orngsqr 30034
Description: In an ordered ring, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Hypotheses
Ref Expression
orngmul.0 𝐵 = (Base‘𝑅)
orngmul.1 = (le‘𝑅)
orngmul.2 0 = (0g𝑅)
orngmul.3 · = (.r𝑅)
Assertion
Ref Expression
orngsqr ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 0 (𝑋 · 𝑋))

Proof of Theorem orngsqr
StepHypRef Expression
1 simpll 807 . . 3 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝑅 ∈ oRing)
2 simplr 809 . . 3 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝑋𝐵)
3 simpr 479 . . 3 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ 0 𝑋) → 0 𝑋)
4 orngmul.0 . . . 4 𝐵 = (Base‘𝑅)
5 orngmul.1 . . . 4 = (le‘𝑅)
6 orngmul.2 . . . 4 0 = (0g𝑅)
7 orngmul.3 . . . 4 · = (.r𝑅)
84, 5, 6, 7orngmul 30033 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑋𝐵0 𝑋)) → 0 (𝑋 · 𝑋))
91, 2, 3, 2, 3, 8syl122anc 1448 . 2 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ 0 𝑋) → 0 (𝑋 · 𝑋))
10 simpll 807 . . . 4 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑅 ∈ oRing)
11 orngring 30030 . . . . . . 7 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
1211ad2antrr 764 . . . . . 6 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑅 ∈ Ring)
13 ringgrp 18673 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
1412, 13syl 17 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑅 ∈ Grp)
15 simplr 809 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑋𝐵)
16 eqid 2724 . . . . . 6 (invg𝑅) = (invg𝑅)
174, 16grpinvcl 17589 . . . . 5 ((𝑅 ∈ Grp ∧ 𝑋𝐵) → ((invg𝑅)‘𝑋) ∈ 𝐵)
1814, 15, 17syl2anc 696 . . . 4 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ((invg𝑅)‘𝑋) ∈ 𝐵)
19 orngogrp 30031 . . . . . . . 8 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
20 isogrp 29932 . . . . . . . . 9 (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
2120simprbi 483 . . . . . . . 8 (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
2219, 21syl 17 . . . . . . 7 (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
2310, 22syl 17 . . . . . 6 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑅 ∈ oMnd)
244, 6grpidcl 17572 . . . . . . 7 (𝑅 ∈ Grp → 0𝐵)
2514, 24syl 17 . . . . . 6 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 0𝐵)
26 simpl 474 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 𝑅 ∈ oRing)
2711, 13, 243syl 18 . . . . . . . . . . . 12 (𝑅 ∈ oRing → 0𝐵)
2826, 27syl 17 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 0𝐵)
29 simpr 479 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 𝑋𝐵)
3026, 28, 293jca 1379 . . . . . . . . . 10 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → (𝑅 ∈ oRing ∧ 0𝐵𝑋𝐵))
31 eqid 2724 . . . . . . . . . . . 12 (lt‘𝑅) = (lt‘𝑅)
325, 31pltle 17083 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ 0𝐵𝑋𝐵) → ( 0 (lt‘𝑅)𝑋0 𝑋))
3332con3dimp 456 . . . . . . . . . 10 (((𝑅 ∈ oRing ∧ 0𝐵𝑋𝐵) ∧ ¬ 0 𝑋) → ¬ 0 (lt‘𝑅)𝑋)
3430, 33sylan 489 . . . . . . . . 9 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ¬ 0 (lt‘𝑅)𝑋)
35 omndtos 29935 . . . . . . . . . . . . 13 (𝑅 ∈ oMnd → 𝑅 ∈ Toset)
3622, 35syl 17 . . . . . . . . . . . 12 (𝑅 ∈ oRing → 𝑅 ∈ Toset)
374, 5, 31tosso 17158 . . . . . . . . . . . . . 14 (𝑅 ∈ Toset → (𝑅 ∈ Toset ↔ ((lt‘𝑅) Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
3837ibi 256 . . . . . . . . . . . . 13 (𝑅 ∈ Toset → ((lt‘𝑅) Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ))
3938simpld 477 . . . . . . . . . . . 12 (𝑅 ∈ Toset → (lt‘𝑅) Or 𝐵)
4010, 36, 393syl 18 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (lt‘𝑅) Or 𝐵)
41 solin 5162 . . . . . . . . . . 11 (((lt‘𝑅) Or 𝐵 ∧ ( 0𝐵𝑋𝐵)) → ( 0 (lt‘𝑅)𝑋0 = 𝑋𝑋(lt‘𝑅) 0 ))
4240, 25, 15, 41syl12anc 1437 . . . . . . . . . 10 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ( 0 (lt‘𝑅)𝑋0 = 𝑋𝑋(lt‘𝑅) 0 ))
43 3orass 1075 . . . . . . . . . 10 (( 0 (lt‘𝑅)𝑋0 = 𝑋𝑋(lt‘𝑅) 0 ) ↔ ( 0 (lt‘𝑅)𝑋 ∨ ( 0 = 𝑋𝑋(lt‘𝑅) 0 )))
4442, 43sylib 208 . . . . . . . . 9 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ( 0 (lt‘𝑅)𝑋 ∨ ( 0 = 𝑋𝑋(lt‘𝑅) 0 )))
45 orel1 396 . . . . . . . . 9 0 (lt‘𝑅)𝑋 → (( 0 (lt‘𝑅)𝑋 ∨ ( 0 = 𝑋𝑋(lt‘𝑅) 0 )) → ( 0 = 𝑋𝑋(lt‘𝑅) 0 )))
4634, 44, 45sylc 65 . . . . . . . 8 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ( 0 = 𝑋𝑋(lt‘𝑅) 0 ))
47 orcom 401 . . . . . . . . 9 (( 0 = 𝑋𝑋(lt‘𝑅) 0 ) ↔ (𝑋(lt‘𝑅) 00 = 𝑋))
48 eqcom 2731 . . . . . . . . . 10 ( 0 = 𝑋𝑋 = 0 )
4948orbi2i 542 . . . . . . . . 9 ((𝑋(lt‘𝑅) 00 = 𝑋) ↔ (𝑋(lt‘𝑅) 0𝑋 = 0 ))
5047, 49bitri 264 . . . . . . . 8 (( 0 = 𝑋𝑋(lt‘𝑅) 0 ) ↔ (𝑋(lt‘𝑅) 0𝑋 = 0 ))
5146, 50sylib 208 . . . . . . 7 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (𝑋(lt‘𝑅) 0𝑋 = 0 ))
52 tospos 29888 . . . . . . . . 9 (𝑅 ∈ Toset → 𝑅 ∈ Poset)
5310, 36, 523syl 18 . . . . . . . 8 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑅 ∈ Poset)
544, 5, 31pleval2 17087 . . . . . . . 8 ((𝑅 ∈ Poset ∧ 𝑋𝐵0𝐵) → (𝑋 0 ↔ (𝑋(lt‘𝑅) 0𝑋 = 0 )))
5553, 15, 25, 54syl3anc 1439 . . . . . . 7 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (𝑋 0 ↔ (𝑋(lt‘𝑅) 0𝑋 = 0 )))
5651, 55mpbird 247 . . . . . 6 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑋 0 )
57 eqid 2724 . . . . . . 7 (+g𝑅) = (+g𝑅)
584, 5, 57omndadd 29936 . . . . . 6 ((𝑅 ∈ oMnd ∧ (𝑋𝐵0𝐵 ∧ ((invg𝑅)‘𝑋) ∈ 𝐵) ∧ 𝑋 0 ) → (𝑋(+g𝑅)((invg𝑅)‘𝑋)) ( 0 (+g𝑅)((invg𝑅)‘𝑋)))
5923, 15, 25, 18, 56, 58syl131anc 1452 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (𝑋(+g𝑅)((invg𝑅)‘𝑋)) ( 0 (+g𝑅)((invg𝑅)‘𝑋)))
604, 57, 6, 16grprinv 17591 . . . . . 6 ((𝑅 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝑅)((invg𝑅)‘𝑋)) = 0 )
6114, 15, 60syl2anc 696 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (𝑋(+g𝑅)((invg𝑅)‘𝑋)) = 0 )
624, 57, 6grplid 17574 . . . . . 6 ((𝑅 ∈ Grp ∧ ((invg𝑅)‘𝑋) ∈ 𝐵) → ( 0 (+g𝑅)((invg𝑅)‘𝑋)) = ((invg𝑅)‘𝑋))
6314, 18, 62syl2anc 696 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ( 0 (+g𝑅)((invg𝑅)‘𝑋)) = ((invg𝑅)‘𝑋))
6459, 61, 633brtr3d 4791 . . . 4 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 0 ((invg𝑅)‘𝑋))
654, 5, 6, 7orngmul 30033 . . . 4 ((𝑅 ∈ oRing ∧ (((invg𝑅)‘𝑋) ∈ 𝐵0 ((invg𝑅)‘𝑋)) ∧ (((invg𝑅)‘𝑋) ∈ 𝐵0 ((invg𝑅)‘𝑋))) → 0 (((invg𝑅)‘𝑋) · ((invg𝑅)‘𝑋)))
6610, 18, 64, 18, 64, 65syl122anc 1448 . . 3 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 0 (((invg𝑅)‘𝑋) · ((invg𝑅)‘𝑋)))
674, 7, 16, 12, 15, 15ringm2neg 18719 . . 3 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (((invg𝑅)‘𝑋) · ((invg𝑅)‘𝑋)) = (𝑋 · 𝑋))
6866, 67breqtrd 4786 . 2 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 0 (𝑋 · 𝑋))
699, 68pm2.61dan 867 1 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 0 (𝑋 · 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3o 1071  w3a 1072   = wceq 1596  wcel 2103  wss 3680   class class class wbr 4760   I cid 5127   Or wor 5138  cres 5220  cfv 6001  (class class class)co 6765  Basecbs 15980  +gcplusg 16064  .rcmulr 16065  lecple 16071  0gc0g 16223  Posetcpo 17062  ltcplt 17063  Tosetctos 17155  Grpcgrp 17544  invgcminusg 17545  Ringcrg 18668  oMndcomnd 29927  oGrpcogrp 29928  oRingcorng 30025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-plusg 16077  df-0g 16225  df-preset 17050  df-poset 17068  df-plt 17080  df-toset 17156  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-grp 17547  df-minusg 17548  df-mgp 18611  df-ur 18623  df-ring 18670  df-omnd 29929  df-ogrp 29930  df-orng 30027
This theorem is referenced by:  orng0le1  30042
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