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Theorem orngsqr 30870
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 765 . . 3 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝑅 ∈ oRing)
2 simplr 767 . . 3 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ 0 𝑋) → 𝑋𝐵)
3 simpr 487 . . 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 30869 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑋𝐵0 𝑋)) → 0 (𝑋 · 𝑋))
91, 2, 3, 2, 3, 8syl122anc 1373 . 2 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ 0 𝑋) → 0 (𝑋 · 𝑋))
10 simpll 765 . . . 4 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑅 ∈ oRing)
11 orngring 30866 . . . . . . 7 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
1211ad2antrr 724 . . . . . 6 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑅 ∈ Ring)
13 ringgrp 19294 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
1412, 13syl 17 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑅 ∈ Grp)
15 simplr 767 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑋𝐵)
16 eqid 2819 . . . . . 6 (invg𝑅) = (invg𝑅)
174, 16grpinvcl 18143 . . . . 5 ((𝑅 ∈ Grp ∧ 𝑋𝐵) → ((invg𝑅)‘𝑋) ∈ 𝐵)
1814, 15, 17syl2anc 586 . . . 4 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ((invg𝑅)‘𝑋) ∈ 𝐵)
19 orngogrp 30867 . . . . . . . 8 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
20 isogrp 30696 . . . . . . . . 9 (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
2120simprbi 499 . . . . . . . 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 18123 . . . . . . 7 (𝑅 ∈ Grp → 0𝐵)
2514, 24syl 17 . . . . . 6 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 0𝐵)
26 simpl 485 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 𝑅 ∈ oRing)
2711, 13, 243syl 18 . . . . . . . . . . . 12 (𝑅 ∈ oRing → 0𝐵)
2826, 27syl 17 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 0𝐵)
29 simpr 487 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 𝑋𝐵)
3026, 28, 293jca 1122 . . . . . . . . . 10 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → (𝑅 ∈ oRing ∧ 0𝐵𝑋𝐵))
31 eqid 2819 . . . . . . . . . . . 12 (lt‘𝑅) = (lt‘𝑅)
325, 31pltle 17563 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ 0𝐵𝑋𝐵) → ( 0 (lt‘𝑅)𝑋0 𝑋))
3332con3dimp 411 . . . . . . . . . 10 (((𝑅 ∈ oRing ∧ 0𝐵𝑋𝐵) ∧ ¬ 0 𝑋) → ¬ 0 (lt‘𝑅)𝑋)
3430, 33sylan 582 . . . . . . . . 9 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ¬ 0 (lt‘𝑅)𝑋)
35 omndtos 30699 . . . . . . . . . . . . 13 (𝑅 ∈ oMnd → 𝑅 ∈ Toset)
3622, 35syl 17 . . . . . . . . . . . 12 (𝑅 ∈ oRing → 𝑅 ∈ Toset)
374, 5, 31tosso 17638 . . . . . . . . . . . . . 14 (𝑅 ∈ Toset → (𝑅 ∈ Toset ↔ ((lt‘𝑅) Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
3837ibi 269 . . . . . . . . . . . . 13 (𝑅 ∈ Toset → ((lt‘𝑅) Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ))
3938simpld 497 . . . . . . . . . . . 12 (𝑅 ∈ Toset → (lt‘𝑅) Or 𝐵)
4010, 36, 393syl 18 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (lt‘𝑅) Or 𝐵)
41 solin 5491 . . . . . . . . . . 11 (((lt‘𝑅) Or 𝐵 ∧ ( 0𝐵𝑋𝐵)) → ( 0 (lt‘𝑅)𝑋0 = 𝑋𝑋(lt‘𝑅) 0 ))
4240, 25, 15, 41syl12anc 834 . . . . . . . . . 10 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ( 0 (lt‘𝑅)𝑋0 = 𝑋𝑋(lt‘𝑅) 0 ))
43 3orass 1084 . . . . . . . . . 10 (( 0 (lt‘𝑅)𝑋0 = 𝑋𝑋(lt‘𝑅) 0 ) ↔ ( 0 (lt‘𝑅)𝑋 ∨ ( 0 = 𝑋𝑋(lt‘𝑅) 0 )))
4442, 43sylib 220 . . . . . . . . 9 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ( 0 (lt‘𝑅)𝑋 ∨ ( 0 = 𝑋𝑋(lt‘𝑅) 0 )))
45 orel1 884 . . . . . . . . 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 866 . . . . . . . . 9 (( 0 = 𝑋𝑋(lt‘𝑅) 0 ) ↔ (𝑋(lt‘𝑅) 00 = 𝑋))
48 eqcom 2826 . . . . . . . . . 10 ( 0 = 𝑋𝑋 = 0 )
4948orbi2i 908 . . . . . . . . 9 ((𝑋(lt‘𝑅) 00 = 𝑋) ↔ (𝑋(lt‘𝑅) 0𝑋 = 0 ))
5047, 49bitri 277 . . . . . . . 8 (( 0 = 𝑋𝑋(lt‘𝑅) 0 ) ↔ (𝑋(lt‘𝑅) 0𝑋 = 0 ))
5146, 50sylib 220 . . . . . . 7 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (𝑋(lt‘𝑅) 0𝑋 = 0 ))
52 tospos 30638 . . . . . . . . 9 (𝑅 ∈ Toset → 𝑅 ∈ Poset)
5310, 36, 523syl 18 . . . . . . . 8 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑅 ∈ Poset)
544, 5, 31pleval2 17567 . . . . . . . 8 ((𝑅 ∈ Poset ∧ 𝑋𝐵0𝐵) → (𝑋 0 ↔ (𝑋(lt‘𝑅) 0𝑋 = 0 )))
5553, 15, 25, 54syl3anc 1365 . . . . . . 7 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (𝑋 0 ↔ (𝑋(lt‘𝑅) 0𝑋 = 0 )))
5651, 55mpbird 259 . . . . . 6 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 𝑋 0 )
57 eqid 2819 . . . . . . 7 (+g𝑅) = (+g𝑅)
584, 5, 57omndadd 30700 . . . . . 6 ((𝑅 ∈ oMnd ∧ (𝑋𝐵0𝐵 ∧ ((invg𝑅)‘𝑋) ∈ 𝐵) ∧ 𝑋 0 ) → (𝑋(+g𝑅)((invg𝑅)‘𝑋)) ( 0 (+g𝑅)((invg𝑅)‘𝑋)))
5923, 15, 25, 18, 56, 58syl131anc 1377 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (𝑋(+g𝑅)((invg𝑅)‘𝑋)) ( 0 (+g𝑅)((invg𝑅)‘𝑋)))
604, 57, 6, 16grprinv 18145 . . . . . 6 ((𝑅 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝑅)((invg𝑅)‘𝑋)) = 0 )
6114, 15, 60syl2anc 586 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (𝑋(+g𝑅)((invg𝑅)‘𝑋)) = 0 )
624, 57, 6grplid 18125 . . . . . 6 ((𝑅 ∈ Grp ∧ ((invg𝑅)‘𝑋) ∈ 𝐵) → ( 0 (+g𝑅)((invg𝑅)‘𝑋)) = ((invg𝑅)‘𝑋))
6314, 18, 62syl2anc 586 . . . . 5 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → ( 0 (+g𝑅)((invg𝑅)‘𝑋)) = ((invg𝑅)‘𝑋))
6459, 61, 633brtr3d 5088 . . . 4 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 0 ((invg𝑅)‘𝑋))
654, 5, 6, 7orngmul 30869 . . . 4 ((𝑅 ∈ oRing ∧ (((invg𝑅)‘𝑋) ∈ 𝐵0 ((invg𝑅)‘𝑋)) ∧ (((invg𝑅)‘𝑋) ∈ 𝐵0 ((invg𝑅)‘𝑋))) → 0 (((invg𝑅)‘𝑋) · ((invg𝑅)‘𝑋)))
6610, 18, 64, 18, 64, 65syl122anc 1373 . . 3 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 0 (((invg𝑅)‘𝑋) · ((invg𝑅)‘𝑋)))
674, 7, 16, 12, 15, 15ringm2neg 19340 . . 3 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → (((invg𝑅)‘𝑋) · ((invg𝑅)‘𝑋)) = (𝑋 · 𝑋))
6866, 67breqtrd 5083 . 2 (((𝑅 ∈ oRing ∧ 𝑋𝐵) ∧ ¬ 0 𝑋) → 0 (𝑋 · 𝑋))
699, 68pm2.61dan 811 1 ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 0 (𝑋 · 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3o 1080  w3a 1081   = wceq 1530  wcel 2107  wss 3934   class class class wbr 5057   I cid 5452   Or wor 5466  cres 5550  cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  lecple 16564  0gc0g 16705  Posetcpo 17542  ltcplt 17543  Tosetctos 17635  Grpcgrp 18095  invgcminusg 18096  Ringcrg 19289  oMndcomnd 30691  oGrpcogrp 30692  oRingcorng 30861
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-0g 16707  df-proset 17530  df-poset 17548  df-plt 17560  df-toset 17636  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-mgp 19232  df-ur 19244  df-ring 19291  df-omnd 30693  df-ogrp 30694  df-orng 30863
This theorem is referenced by:  orng0le1  30878
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