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Theorem suborng 30149
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 471 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Ring)
2 ringgrp 18759 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Grp)
32adantl 467 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Grp)
4 orngogrp 30135 . . . . 5 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5 isogrp 30036 . . . . . 6 (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
65simprbi 478 . . . . 5 (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
74, 6syl 17 . . . 4 (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
8 ringmnd 18763 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Mnd)
9 submomnd 30044 . . . 4 ((𝑅 ∈ oMnd ∧ (𝑅s 𝐴) ∈ Mnd) → (𝑅s 𝐴) ∈ oMnd)
107, 8, 9syl2an 575 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oMnd)
11 isogrp 30036 . . 3 ((𝑅s 𝐴) ∈ oGrp ↔ ((𝑅s 𝐴) ∈ Grp ∧ (𝑅s 𝐴) ∈ oMnd))
123, 10, 11sylanbrc 564 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oGrp)
13 simp-4l 760 . . . . . . 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 16132 . . . . . . . . . . . . . . 15 Rel dom ↾s
1514ovprc2 6829 . . . . . . . . . . . . . 14 𝐴 ∈ V → (𝑅s 𝐴) = ∅)
1615fveq2d 6336 . . . . . . . . . . . . 13 𝐴 ∈ V → (Base‘(𝑅s 𝐴)) = (Base‘∅))
1716adantl 467 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = (Base‘∅))
18 base0 16118 . . . . . . . . . . . 12 ∅ = (Base‘∅)
1917, 18syl6eqr 2822 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = ∅)
20 eqid 2770 . . . . . . . . . . . . . . 15 (Base‘(𝑅s 𝐴)) = (Base‘(𝑅s 𝐴))
21 eqid 2770 . . . . . . . . . . . . . . 15 (1r‘(𝑅s 𝐴)) = (1r‘(𝑅s 𝐴))
2220, 21ringidcl 18775 . . . . . . . . . . . . . 14 ((𝑅s 𝐴) ∈ Ring → (1r‘(𝑅s 𝐴)) ∈ (Base‘(𝑅s 𝐴)))
23 ne0i 4067 . . . . . . . . . . . . . 14 ((1r‘(𝑅s 𝐴)) ∈ (Base‘(𝑅s 𝐴)) → (Base‘(𝑅s 𝐴)) ≠ ∅)
2422, 23syl 17 . . . . . . . . . . . . 13 ((𝑅s 𝐴) ∈ Ring → (Base‘(𝑅s 𝐴)) ≠ ∅)
2524ad2antlr 698 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) ≠ ∅)
2625neneqd 2947 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅s 𝐴)) = ∅)
2719, 26condan 801 . . . . . . . . . 10 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝐴 ∈ V)
28 eqid 2770 . . . . . . . . . . . 12 (𝑅s 𝐴) = (𝑅s 𝐴)
29 eqid 2770 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
3028, 29ressbas 16136 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅s 𝐴)))
31 inss2 3980 . . . . . . . . . . 11 (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
3230, 31syl6eqssr 3803 . . . . . . . . . 10 (𝐴 ∈ V → (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅))
3327, 32syl 17 . . . . . . . . 9 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅))
3433ad3antrrr 701 . . . . . . . 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))
35 simpllr 752 . . . . . . . 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)))
3634, 35sseldd 3751 . . . . . . 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))
37 simprl 746 . . . . . . . 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)
38 orngring 30134 . . . . . . . . . . . . . . . 16 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
39 ringgrp 18759 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
4038, 39syl 17 . . . . . . . . . . . . . . 15 (𝑅 ∈ oRing → 𝑅 ∈ Grp)
4140adantr 466 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝑅 ∈ Grp)
4229ressinbas 16142 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s 𝐴) = (𝑅s (𝐴 ∩ (Base‘𝑅))))
4330oveq2d 6808 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s (𝐴 ∩ (Base‘𝑅))) = (𝑅s (Base‘(𝑅s 𝐴))))
4442, 43eqtrd 2804 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V → (𝑅s 𝐴) = (𝑅s (Base‘(𝑅s 𝐴))))
4527, 44syl 17 . . . . . . . . . . . . . . 15 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) = (𝑅s (Base‘(𝑅s 𝐴))))
4645, 3eqeltrrd 2850 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s (Base‘(𝑅s 𝐴))) ∈ Grp)
4729issubg 17801 . . . . . . . . . . . . . 14 ((Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅) ∧ (𝑅s (Base‘(𝑅s 𝐴))) ∈ Grp))
4841, 33, 46, 47syl3anbrc 1427 . . . . . . . . . . . . 13 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅))
49 eqid 2770 . . . . . . . . . . . . . 14 (𝑅s (Base‘(𝑅s 𝐴))) = (𝑅s (Base‘(𝑅s 𝐴)))
50 eqid 2770 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5149, 50subg0 17807 . . . . . . . . . . . . 13 ((Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅) → (0g𝑅) = (0g‘(𝑅s (Base‘(𝑅s 𝐴)))))
5248, 51syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g𝑅) = (0g‘(𝑅s (Base‘(𝑅s 𝐴)))))
5345fveq2d 6336 . . . . . . . . . . . 12 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s (Base‘(𝑅s 𝐴)))))
5452, 53eqtr4d 2807 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g𝑅) = (0g‘(𝑅s 𝐴)))
5554ad2antrr 697 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → (0g𝑅) = (0g‘(𝑅s 𝐴)))
5627ad2antrr 697 . . . . . . . . . . 11 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝐴 ∈ V)
57 eqid 2770 . . . . . . . . . . . 12 (le‘𝑅) = (le‘𝑅)
5828, 57ressle 16266 . . . . . . . . . . 11 (𝐴 ∈ V → (le‘𝑅) = (le‘(𝑅s 𝐴)))
5956, 58syl 17 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → (le‘𝑅) = (le‘(𝑅s 𝐴)))
60 eqidd 2771 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑎 = 𝑎)
6155, 59, 60breq123d 4798 . . . . . . . . 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))
6261adantr 466 . . . . . . . 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))
6337, 62mpbird 247 . . . . . . 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)
64 simplr 744 . . . . . . . 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)))
6534, 64sseldd 3751 . . . . . . 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))
66 simprr 748 . . . . . . . 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)
67 eqidd 2771 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑏 = 𝑏)
6855, 59, 67breq123d 4798 . . . . . . . . 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))
6968adantr 466 . . . . . . . 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))
7066, 69mpbird 247 . . . . . . 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)
71 eqid 2770 . . . . . . . 8 (.r𝑅) = (.r𝑅)
7229, 57, 50, 71orngmul 30137 . . . . . . 7 ((𝑅 ∈ oRing ∧ (𝑎 ∈ (Base‘𝑅) ∧ (0g𝑅)(le‘𝑅)𝑎) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (0g𝑅)(le‘𝑅)𝑏)) → (0g𝑅)(le‘𝑅)(𝑎(.r𝑅)𝑏))
7313, 36, 63, 65, 70, 72syl122anc 1484 . . . . . 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))
7455adantr 466 . . . . . . 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)))
7559adantr 466 . . . . . . 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)))
7656adantr 466 . . . . . . . . 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)
7728, 71ressmulr 16213 . . . . . . . . 9 (𝐴 ∈ V → (.r𝑅) = (.r‘(𝑅s 𝐴)))
7876, 77syl 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)))
7978oveqd 6809 . . . . . . 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))
8074, 75, 79breq123d 4798 . . . . . 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)))
8173, 80mpbid 222 . . . . 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))
8281ex 397 . . . 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)))
8382anasss 457 . . 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)))
8483ralrimivva 3119 . 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)))
85 eqid 2770 . . 3 (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s 𝐴))
86 eqid 2770 . . 3 (.r‘(𝑅s 𝐴)) = (.r‘(𝑅s 𝐴))
87 eqid 2770 . . 3 (le‘(𝑅s 𝐴)) = (le‘(𝑅s 𝐴))
8820, 85, 86, 87isorng 30133 . 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))))
891, 12, 84, 88syl3anbrc 1427 1 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wne 2942  wral 3060  Vcvv 3349  cin 3720  wss 3721  c0 4061   class class class wbr 4784  cfv 6031  (class class class)co 6792  Basecbs 16063  s cress 16064  .rcmulr 16149  lecple 16155  0gc0g 16307  Mndcmnd 17501  Grpcgrp 17629  SubGrpcsubg 17795  1rcur 18708  Ringcrg 18754  oMndcomnd 30031  oGrpcogrp 30032  oRingcorng 30129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-dec 11695  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-ple 16168  df-0g 16309  df-poset 17153  df-toset 17241  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-subg 17798  df-mgp 18697  df-ur 18709  df-ring 18756  df-omnd 30033  df-ogrp 30034  df-orng 30131
This theorem is referenced by:  subofld  30150
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