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Theorem suborng 30895
 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 488 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Ring)
2 ringgrp 19280 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Grp)
32adantl 485 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Grp)
4 orngogrp 30881 . . . . 5 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5 isogrp 30710 . . . . . 6 (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
65simprbi 500 . . . . 5 (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
74, 6syl 17 . . . 4 (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
8 ringmnd 19284 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Mnd)
9 submomnd 30718 . . . 4 ((𝑅 ∈ oMnd ∧ (𝑅s 𝐴) ∈ Mnd) → (𝑅s 𝐴) ∈ oMnd)
107, 8, 9syl2an 598 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oMnd)
11 isogrp 30710 . . 3 ((𝑅s 𝐴) ∈ oGrp ↔ ((𝑅s 𝐴) ∈ Grp ∧ (𝑅s 𝐴) ∈ oMnd))
123, 10, 11sylanbrc 586 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oGrp)
13 simp-4l 782 . . . . . . 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 16528 . . . . . . . . . . . . . . 15 Rel dom ↾s
1514ovprc2 7170 . . . . . . . . . . . . . 14 𝐴 ∈ V → (𝑅s 𝐴) = ∅)
1615fveq2d 6647 . . . . . . . . . . . . 13 𝐴 ∈ V → (Base‘(𝑅s 𝐴)) = (Base‘∅))
1716adantl 485 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = (Base‘∅))
18 base0 16514 . . . . . . . . . . . 12 ∅ = (Base‘∅)
1917, 18syl6eqr 2874 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = ∅)
20 eqid 2821 . . . . . . . . . . . . . . 15 (Base‘(𝑅s 𝐴)) = (Base‘(𝑅s 𝐴))
21 eqid 2821 . . . . . . . . . . . . . . 15 (1r‘(𝑅s 𝐴)) = (1r‘(𝑅s 𝐴))
2220, 21ringidcl 19296 . . . . . . . . . . . . . 14 ((𝑅s 𝐴) ∈ Ring → (1r‘(𝑅s 𝐴)) ∈ (Base‘(𝑅s 𝐴)))
2322ne0d 4274 . . . . . . . . . . . . 13 ((𝑅s 𝐴) ∈ Ring → (Base‘(𝑅s 𝐴)) ≠ ∅)
2423ad2antlr 726 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) ≠ ∅)
2524neneqd 3012 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅s 𝐴)) = ∅)
2619, 25condan 817 . . . . . . . . . 10 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝐴 ∈ V)
27 eqid 2821 . . . . . . . . . . . 12 (𝑅s 𝐴) = (𝑅s 𝐴)
28 eqid 2821 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
2927, 28ressbas 16532 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅s 𝐴)))
30 inss2 4181 . . . . . . . . . . 11 (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
3129, 30eqsstrrdi 3998 . . . . . . . . . 10 (𝐴 ∈ V → (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅))
3226, 31syl 17 . . . . . . . . 9 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅))
3332ad3antrrr 729 . . . . . . . 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 775 . . . . . . . 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 3944 . . . . . . 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 770 . . . . . . . 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 30880 . . . . . . . . . . . . . . . 16 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38 ringgrp 19280 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3937, 38syl 17 . . . . . . . . . . . . . . 15 (𝑅 ∈ oRing → 𝑅 ∈ Grp)
4039adantr 484 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝑅 ∈ Grp)
4128ressinbas 16538 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s 𝐴) = (𝑅s (𝐴 ∩ (Base‘𝑅))))
4229oveq2d 7146 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s (𝐴 ∩ (Base‘𝑅))) = (𝑅s (Base‘(𝑅s 𝐴))))
4341, 42eqtrd 2856 . . . . . . . . . . . . . . . 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 2913 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s (Base‘(𝑅s 𝐴))) ∈ Grp)
4628issubg 18257 . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . 14 (𝑅s (Base‘(𝑅s 𝐴))) = (𝑅s (Base‘(𝑅s 𝐴)))
49 eqid 2821 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5048, 49subg0 18263 . . . . . . . . . . . . 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 6647 . . . . . . . . . . . 12 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s (Base‘(𝑅s 𝐴)))))
5351, 52eqtr4d 2859 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g𝑅) = (0g‘(𝑅s 𝐴)))
5453ad2antrr 725 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → (0g𝑅) = (0g‘(𝑅s 𝐴)))
5526ad2antrr 725 . . . . . . . . . . 11 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝐴 ∈ V)
56 eqid 2821 . . . . . . . . . . . 12 (le‘𝑅) = (le‘𝑅)
5727, 56ressle 16650 . . . . . . . . . . 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 2822 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑎 = 𝑎)
6054, 58, 59breq123d 5053 . . . . . . . . 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 484 . . . . . . . 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 260 . . . . . . 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 768 . . . . . . . 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 3944 . . . . . . 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 772 . . . . . . . 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 2822 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑏 = 𝑏)
6754, 58, 66breq123d 5053 . . . . . . . . 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 484 . . . . . . . 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 260 . . . . . . 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 2821 . . . . . . . 8 (.r𝑅) = (.r𝑅)
7128, 56, 49, 70orngmul 30883 . . . . . . 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 484 . . . . . . 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 484 . . . . . . 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 484 . . . . . . . . 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 16603 . . . . . . . . 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 7147 . . . . . . 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 5053 . . . . . 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 235 . . . . 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 416 . . . 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 470 . . 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 3179 . 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 2821 . . 3 (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s 𝐴))
85 eqid 2821 . . 3 (.r‘(𝑅s 𝐴)) = (.r‘(𝑅s 𝐴))
86 eqid 2821 . . 3 (le‘(𝑅s 𝐴)) = (le‘(𝑅s 𝐴))
8720, 84, 85, 86isorng 30879 . 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ≠ wne 3007  ∀wral 3126  Vcvv 3471   ∩ cin 3909   ⊆ wss 3910  ∅c0 4266   class class class wbr 5039  ‘cfv 6328  (class class class)co 7130  Basecbs 16461   ↾s cress 16462  .rcmulr 16544  lecple 16550  0gc0g 16691  Mndcmnd 17889  Grpcgrp 18081  SubGrpcsubg 18251  1rcur 19229  Ringcrg 19275  oMndcomnd 30705  oGrpcogrp 30706  oRingcorng 30875 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-dec 12077  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-mulr 16557  df-ple 16563  df-0g 16693  df-poset 17534  df-toset 17622  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-grp 18084  df-subg 18254  df-mgp 19218  df-ur 19230  df-ring 19277  df-omnd 30707  df-ogrp 30708  df-orng 30877 This theorem is referenced by:  subofld  30896
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