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Theorem suborng 33129
Description: Every subring of an ordered ring is also an ordered ring. (Contributed by Thierry Arnoux, 21-Jan-2018.)
Assertion
Ref Expression
suborng ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oRing)

Proof of Theorem suborng
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 483 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Ring)
2 ringgrp 20190 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Grp)
32adantl 480 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Grp)
4 orngogrp 33115 . . . . 5 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5 isogrp 32872 . . . . . 6 (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
65simprbi 495 . . . . 5 (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
74, 6syl 17 . . . 4 (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
8 ringmnd 20195 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Mnd)
9 submomnd 32880 . . . 4 ((𝑅 ∈ oMnd ∧ (𝑅s 𝐴) ∈ Mnd) → (𝑅s 𝐴) ∈ oMnd)
107, 8, 9syl2an 594 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oMnd)
11 isogrp 32872 . . 3 ((𝑅s 𝐴) ∈ oGrp ↔ ((𝑅s 𝐴) ∈ Grp ∧ (𝑅s 𝐴) ∈ oMnd))
123, 10, 11sylanbrc 581 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oGrp)
13 simp-4l 781 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → 𝑅 ∈ oRing)
14 reldmress 17214 . . . . . . . . . . . . . . 15 Rel dom ↾s
1514ovprc2 7459 . . . . . . . . . . . . . 14 𝐴 ∈ V → (𝑅s 𝐴) = ∅)
1615fveq2d 6900 . . . . . . . . . . . . 13 𝐴 ∈ V → (Base‘(𝑅s 𝐴)) = (Base‘∅))
1716adantl 480 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = (Base‘∅))
18 base0 17188 . . . . . . . . . . . 12 ∅ = (Base‘∅)
1917, 18eqtr4di 2783 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = ∅)
20 eqid 2725 . . . . . . . . . . . . . . 15 (Base‘(𝑅s 𝐴)) = (Base‘(𝑅s 𝐴))
21 eqid 2725 . . . . . . . . . . . . . . 15 (1r‘(𝑅s 𝐴)) = (1r‘(𝑅s 𝐴))
2220, 21ringidcl 20214 . . . . . . . . . . . . . 14 ((𝑅s 𝐴) ∈ Ring → (1r‘(𝑅s 𝐴)) ∈ (Base‘(𝑅s 𝐴)))
2322ne0d 4335 . . . . . . . . . . . . 13 ((𝑅s 𝐴) ∈ Ring → (Base‘(𝑅s 𝐴)) ≠ ∅)
2423ad2antlr 725 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) ≠ ∅)
2524neneqd 2934 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅s 𝐴)) = ∅)
2619, 25condan 816 . . . . . . . . . 10 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝐴 ∈ V)
27 eqid 2725 . . . . . . . . . . . 12 (𝑅s 𝐴) = (𝑅s 𝐴)
28 eqid 2725 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
2927, 28ressbas 17218 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅s 𝐴)))
30 inss2 4228 . . . . . . . . . . 11 (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
3129, 30eqsstrrdi 4032 . . . . . . . . . 10 (𝐴 ∈ V → (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅))
3226, 31syl 17 . . . . . . . . 9 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅))
3332ad3antrrr 728 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅))
34 simpllr 774 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → 𝑎 ∈ (Base‘(𝑅s 𝐴)))
3533, 34sseldd 3977 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → 𝑎 ∈ (Base‘𝑅))
36 simprl 769 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎)
37 orngring 33114 . . . . . . . . . . . . . . . 16 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38 ringgrp 20190 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3937, 38syl 17 . . . . . . . . . . . . . . 15 (𝑅 ∈ oRing → 𝑅 ∈ Grp)
4039adantr 479 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝑅 ∈ Grp)
4128ressinbas 17229 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s 𝐴) = (𝑅s (𝐴 ∩ (Base‘𝑅))))
4229oveq2d 7435 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s (𝐴 ∩ (Base‘𝑅))) = (𝑅s (Base‘(𝑅s 𝐴))))
4341, 42eqtrd 2765 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V → (𝑅s 𝐴) = (𝑅s (Base‘(𝑅s 𝐴))))
4426, 43syl 17 . . . . . . . . . . . . . . 15 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) = (𝑅s (Base‘(𝑅s 𝐴))))
4544, 3eqeltrrd 2826 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s (Base‘(𝑅s 𝐴))) ∈ Grp)
4628issubg 19089 . . . . . . . . . . . . . 14 ((Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅) ∧ (𝑅s (Base‘(𝑅s 𝐴))) ∈ Grp))
4740, 32, 45, 46syl3anbrc 1340 . . . . . . . . . . . . 13 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅))
48 eqid 2725 . . . . . . . . . . . . . 14 (𝑅s (Base‘(𝑅s 𝐴))) = (𝑅s (Base‘(𝑅s 𝐴)))
49 eqid 2725 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5048, 49subg0 19095 . . . . . . . . . . . . 13 ((Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅) → (0g𝑅) = (0g‘(𝑅s (Base‘(𝑅s 𝐴)))))
5147, 50syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g𝑅) = (0g‘(𝑅s (Base‘(𝑅s 𝐴)))))
5244fveq2d 6900 . . . . . . . . . . . 12 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s (Base‘(𝑅s 𝐴)))))
5351, 52eqtr4d 2768 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g𝑅) = (0g‘(𝑅s 𝐴)))
5453ad2antrr 724 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → (0g𝑅) = (0g‘(𝑅s 𝐴)))
5526ad2antrr 724 . . . . . . . . . . 11 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝐴 ∈ V)
56 eqid 2725 . . . . . . . . . . . 12 (le‘𝑅) = (le‘𝑅)
5727, 56ressle 17364 . . . . . . . . . . 11 (𝐴 ∈ V → (le‘𝑅) = (le‘(𝑅s 𝐴)))
5855, 57syl 17 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → (le‘𝑅) = (le‘(𝑅s 𝐴)))
59 eqidd 2726 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑎 = 𝑎)
6054, 58, 59breq123d 5163 . . . . . . . . 9 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → ((0g𝑅)(le‘𝑅)𝑎 ↔ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎))
6160adantr 479 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → ((0g𝑅)(le‘𝑅)𝑎 ↔ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎))
6236, 61mpbird 256 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (0g𝑅)(le‘𝑅)𝑎)
63 simplr 767 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → 𝑏 ∈ (Base‘(𝑅s 𝐴)))
6433, 63sseldd 3977 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → 𝑏 ∈ (Base‘𝑅))
65 simprr 771 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)
66 eqidd 2726 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑏 = 𝑏)
6754, 58, 66breq123d 5163 . . . . . . . . 9 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → ((0g𝑅)(le‘𝑅)𝑏 ↔ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏))
6867adantr 479 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → ((0g𝑅)(le‘𝑅)𝑏 ↔ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏))
6965, 68mpbird 256 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (0g𝑅)(le‘𝑅)𝑏)
70 eqid 2725 . . . . . . . 8 (.r𝑅) = (.r𝑅)
7128, 56, 49, 70orngmul 33117 . . . . . . 7 ((𝑅 ∈ oRing ∧ (𝑎 ∈ (Base‘𝑅) ∧ (0g𝑅)(le‘𝑅)𝑎) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (0g𝑅)(le‘𝑅)𝑏)) → (0g𝑅)(le‘𝑅)(𝑎(.r𝑅)𝑏))
7213, 35, 62, 64, 69, 71syl122anc 1376 . . . . . 6 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (0g𝑅)(le‘𝑅)(𝑎(.r𝑅)𝑏))
7354adantr 479 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (0g𝑅) = (0g‘(𝑅s 𝐴)))
7458adantr 479 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (le‘𝑅) = (le‘(𝑅s 𝐴)))
7555adantr 479 . . . . . . . . 9 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → 𝐴 ∈ V)
7627, 70ressmulr 17291 . . . . . . . . 9 (𝐴 ∈ V → (.r𝑅) = (.r‘(𝑅s 𝐴)))
7775, 76syl 17 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (.r𝑅) = (.r‘(𝑅s 𝐴)))
7877oveqd 7436 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (𝑎(.r𝑅)𝑏) = (𝑎(.r‘(𝑅s 𝐴))𝑏))
7973, 74, 78breq123d 5163 . . . . . 6 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → ((0g𝑅)(le‘𝑅)(𝑎(.r𝑅)𝑏) ↔ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))(𝑎(.r‘(𝑅s 𝐴))𝑏)))
8072, 79mpbid 231 . . . . 5 (((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) ∧ ((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏)) → (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))(𝑎(.r‘(𝑅s 𝐴))𝑏))
8180ex 411 . . . 4 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → (((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏) → (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))(𝑎(.r‘(𝑅s 𝐴))𝑏)))
8281anasss 465 . . 3 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝑎 ∈ (Base‘(𝑅s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴)))) → (((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏) → (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))(𝑎(.r‘(𝑅s 𝐴))𝑏)))
8382ralrimivva 3190 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → ∀𝑎 ∈ (Base‘(𝑅s 𝐴))∀𝑏 ∈ (Base‘(𝑅s 𝐴))(((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏) → (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))(𝑎(.r‘(𝑅s 𝐴))𝑏)))
84 eqid 2725 . . 3 (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s 𝐴))
85 eqid 2725 . . 3 (.r‘(𝑅s 𝐴)) = (.r‘(𝑅s 𝐴))
86 eqid 2725 . . 3 (le‘(𝑅s 𝐴)) = (le‘(𝑅s 𝐴))
8720, 84, 85, 86isorng 33113 . 2 ((𝑅s 𝐴) ∈ oRing ↔ ((𝑅s 𝐴) ∈ Ring ∧ (𝑅s 𝐴) ∈ oGrp ∧ ∀𝑎 ∈ (Base‘(𝑅s 𝐴))∀𝑏 ∈ (Base‘(𝑅s 𝐴))(((0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑎 ∧ (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))𝑏) → (0g‘(𝑅s 𝐴))(le‘(𝑅s 𝐴))(𝑎(.r‘(𝑅s 𝐴))𝑏))))
881, 12, 83, 87syl3anbrc 1340 1 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  Vcvv 3461  cin 3943  wss 3944  c0 4322   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  s cress 17212  .rcmulr 17237  lecple 17243  0gc0g 17424  Mndcmnd 18697  Grpcgrp 18898  SubGrpcsubg 19083  1rcur 20133  Ringcrg 20185  oMndcomnd 32867  oGrpcogrp 32868  oRingcorng 33109
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-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
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 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-dec 12711  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-ple 17256  df-0g 17426  df-poset 18308  df-toset 18412  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-subg 19086  df-mgp 20087  df-ur 20134  df-ring 20187  df-omnd 32869  df-ogrp 32870  df-orng 33111
This theorem is referenced by:  subofld  33130
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