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Theorem suborng 32110
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 485 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Ring)
2 ringgrp 19969 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Grp)
32adantl 482 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Grp)
4 orngogrp 32096 . . . . 5 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5 isogrp 31910 . . . . . 6 (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
65simprbi 497 . . . . 5 (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
74, 6syl 17 . . . 4 (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
8 ringmnd 19974 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Mnd)
9 submomnd 31918 . . . 4 ((𝑅 ∈ oMnd ∧ (𝑅s 𝐴) ∈ Mnd) → (𝑅s 𝐴) ∈ oMnd)
107, 8, 9syl2an 596 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oMnd)
11 isogrp 31910 . . 3 ((𝑅s 𝐴) ∈ oGrp ↔ ((𝑅s 𝐴) ∈ Grp ∧ (𝑅s 𝐴) ∈ oMnd))
123, 10, 11sylanbrc 583 . 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 17114 . . . . . . . . . . . . . . 15 Rel dom ↾s
1514ovprc2 7397 . . . . . . . . . . . . . 14 𝐴 ∈ V → (𝑅s 𝐴) = ∅)
1615fveq2d 6846 . . . . . . . . . . . . 13 𝐴 ∈ V → (Base‘(𝑅s 𝐴)) = (Base‘∅))
1716adantl 482 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = (Base‘∅))
18 base0 17088 . . . . . . . . . . . 12 ∅ = (Base‘∅)
1917, 18eqtr4di 2794 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = ∅)
20 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘(𝑅s 𝐴)) = (Base‘(𝑅s 𝐴))
21 eqid 2736 . . . . . . . . . . . . . . 15 (1r‘(𝑅s 𝐴)) = (1r‘(𝑅s 𝐴))
2220, 21ringidcl 19989 . . . . . . . . . . . . . 14 ((𝑅s 𝐴) ∈ Ring → (1r‘(𝑅s 𝐴)) ∈ (Base‘(𝑅s 𝐴)))
2322ne0d 4295 . . . . . . . . . . . . 13 ((𝑅s 𝐴) ∈ Ring → (Base‘(𝑅s 𝐴)) ≠ ∅)
2423ad2antlr 725 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) ≠ ∅)
2524neneqd 2948 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅s 𝐴)) = ∅)
2619, 25condan 816 . . . . . . . . . 10 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝐴 ∈ V)
27 eqid 2736 . . . . . . . . . . . 12 (𝑅s 𝐴) = (𝑅s 𝐴)
28 eqid 2736 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
2927, 28ressbas 17118 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅s 𝐴)))
30 inss2 4189 . . . . . . . . . . 11 (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
3129, 30eqsstrrdi 3999 . . . . . . . . . 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 3945 . . . . . . 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 32095 . . . . . . . . . . . . . . . 16 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38 ringgrp 19969 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3937, 38syl 17 . . . . . . . . . . . . . . 15 (𝑅 ∈ oRing → 𝑅 ∈ Grp)
4039adantr 481 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝑅 ∈ Grp)
4128ressinbas 17126 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s 𝐴) = (𝑅s (𝐴 ∩ (Base‘𝑅))))
4229oveq2d 7373 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s (𝐴 ∩ (Base‘𝑅))) = (𝑅s (Base‘(𝑅s 𝐴))))
4341, 42eqtrd 2776 . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s (Base‘(𝑅s 𝐴))) ∈ Grp)
4628issubg 18928 . . . . . . . . . . . . . 14 ((Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅) ∧ (𝑅s (Base‘(𝑅s 𝐴))) ∈ Grp))
4740, 32, 45, 46syl3anbrc 1343 . . . . . . . . . . . . 13 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅))
48 eqid 2736 . . . . . . . . . . . . . 14 (𝑅s (Base‘(𝑅s 𝐴))) = (𝑅s (Base‘(𝑅s 𝐴)))
49 eqid 2736 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5048, 49subg0 18934 . . . . . . . . . . . . 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 6846 . . . . . . . . . . . 12 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s (Base‘(𝑅s 𝐴)))))
5351, 52eqtr4d 2779 . . . . . . . . . . 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 2736 . . . . . . . . . . . 12 (le‘𝑅) = (le‘𝑅)
5727, 56ressle 17261 . . . . . . . . . . 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 2737 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑎 = 𝑎)
6054, 58, 59breq123d 5119 . . . . . . . . 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 481 . . . . . . . 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 3945 . . . . . . 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 2737 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑏 = 𝑏)
6754, 58, 66breq123d 5119 . . . . . . . . 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 481 . . . . . . . 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 2736 . . . . . . . 8 (.r𝑅) = (.r𝑅)
7128, 56, 49, 70orngmul 32098 . . . . . . 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 1379 . . . . . 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 481 . . . . . . 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 481 . . . . . . 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 481 . . . . . . . . 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 17188 . . . . . . . . 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 7374 . . . . . . 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 5119 . . . . . 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 413 . . . 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 467 . . 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 3197 . 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 2736 . . 3 (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s 𝐴))
85 eqid 2736 . . 3 (.r‘(𝑅s 𝐴)) = (.r‘(𝑅s 𝐴))
86 eqid 2736 . . 3 (le‘(𝑅s 𝐴)) = (le‘(𝑅s 𝐴))
8720, 84, 85, 86isorng 32094 . 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 1343 1 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  cin 3909  wss 3910  c0 4282   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  s cress 17112  .rcmulr 17134  lecple 17140  0gc0g 17321  Mndcmnd 18556  Grpcgrp 18748  SubGrpcsubg 18922  1rcur 19913  Ringcrg 19964  oMndcomnd 31905  oGrpcogrp 31906  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-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-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-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-dec 12619  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-ple 17153  df-0g 17323  df-poset 18202  df-toset 18306  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-subg 18925  df-mgp 19897  df-ur 19914  df-ring 19966  df-omnd 31907  df-ogrp 31908  df-orng 32092
This theorem is referenced by:  subofld  32111
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