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Theorem suborng 29146
 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 476 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Ring)
2 ringgrp 18375 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Grp)
32adantl 481 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ Grp)
4 orngogrp 29132 . . . . 5 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5 isogrp 29033 . . . . . 6 (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
65simprbi 479 . . . . 5 (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
74, 6syl 17 . . . 4 (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
8 ringmnd 18379 . . . 4 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Mnd)
9 submomnd 29041 . . . 4 ((𝑅 ∈ oMnd ∧ (𝑅s 𝐴) ∈ Mnd) → (𝑅s 𝐴) ∈ oMnd)
107, 8, 9syl2an 493 . . 3 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oMnd)
11 isogrp 29033 . . 3 ((𝑅s 𝐴) ∈ oGrp ↔ ((𝑅s 𝐴) ∈ Grp ∧ (𝑅s 𝐴) ∈ oMnd))
123, 10, 11sylanbrc 695 . 2 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oGrp)
13 simp-4l 802 . . . . . . 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 15753 . . . . . . . . . . . . . . 15 Rel dom ↾s
1514ovprc2 6583 . . . . . . . . . . . . . 14 𝐴 ∈ V → (𝑅s 𝐴) = ∅)
1615fveq2d 6107 . . . . . . . . . . . . 13 𝐴 ∈ V → (Base‘(𝑅s 𝐴)) = (Base‘∅))
1716adantl 481 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = (Base‘∅))
18 base0 15740 . . . . . . . . . . . 12 ∅ = (Base‘∅)
1917, 18syl6eqr 2662 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) = ∅)
20 eqid 2610 . . . . . . . . . . . . . . 15 (Base‘(𝑅s 𝐴)) = (Base‘(𝑅s 𝐴))
21 eqid 2610 . . . . . . . . . . . . . . 15 (1r‘(𝑅s 𝐴)) = (1r‘(𝑅s 𝐴))
2220, 21ringidcl 18391 . . . . . . . . . . . . . 14 ((𝑅s 𝐴) ∈ Ring → (1r‘(𝑅s 𝐴)) ∈ (Base‘(𝑅s 𝐴)))
23 ne0i 3880 . . . . . . . . . . . . . 14 ((1r‘(𝑅s 𝐴)) ∈ (Base‘(𝑅s 𝐴)) → (Base‘(𝑅s 𝐴)) ≠ ∅)
2422, 23syl 17 . . . . . . . . . . . . 13 ((𝑅s 𝐴) ∈ Ring → (Base‘(𝑅s 𝐴)) ≠ ∅)
2524ad2antlr 759 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅s 𝐴)) ≠ ∅)
2625neneqd 2787 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅s 𝐴)) = ∅)
2719, 26condan 831 . . . . . . . . . 10 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝐴 ∈ V)
28 eqid 2610 . . . . . . . . . . . 12 (𝑅s 𝐴) = (𝑅s 𝐴)
29 eqid 2610 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
3028, 29ressbas 15757 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅s 𝐴)))
31 inss2 3796 . . . . . . . . . . 11 (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
3230, 31syl6eqssr 3619 . . . . . . . . . 10 (𝐴 ∈ V → (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅))
3327, 32syl 17 . . . . . . . . 9 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅))
3433ad3antrrr 762 . . . . . . . 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 795 . . . . . . . 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 3569 . . . . . . 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 790 . . . . . . . 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 29131 . . . . . . . . . . . . . . . 16 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
39 ringgrp 18375 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
4038, 39syl 17 . . . . . . . . . . . . . . 15 (𝑅 ∈ oRing → 𝑅 ∈ Grp)
4140adantr 480 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → 𝑅 ∈ Grp)
4229ressinbas 15763 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s 𝐴) = (𝑅s (𝐴 ∩ (Base‘𝑅))))
4330oveq2d 6565 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝑅s (𝐴 ∩ (Base‘𝑅))) = (𝑅s (Base‘(𝑅s 𝐴))))
4442, 43eqtrd 2644 . . . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s (Base‘(𝑅s 𝐴))) ∈ Grp)
4729issubg 17417 . . . . . . . . . . . . . 14 ((Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ (Base‘(𝑅s 𝐴)) ⊆ (Base‘𝑅) ∧ (𝑅s (Base‘(𝑅s 𝐴))) ∈ Grp))
4841, 33, 46, 47syl3anbrc 1239 . . . . . . . . . . . . 13 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (Base‘(𝑅s 𝐴)) ∈ (SubGrp‘𝑅))
49 eqid 2610 . . . . . . . . . . . . . 14 (𝑅s (Base‘(𝑅s 𝐴))) = (𝑅s (Base‘(𝑅s 𝐴)))
50 eqid 2610 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5149, 50subg0 17423 . . . . . . . . . . . . 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 6107 . . . . . . . . . . . 12 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s (Base‘(𝑅s 𝐴)))))
5452, 53eqtr4d 2647 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (0g𝑅) = (0g‘(𝑅s 𝐴)))
5554ad2antrr 758 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → (0g𝑅) = (0g‘(𝑅s 𝐴)))
5627ad2antrr 758 . . . . . . . . . . 11 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝐴 ∈ V)
57 eqid 2610 . . . . . . . . . . . 12 (le‘𝑅) = (le‘𝑅)
5828, 57ressle 15882 . . . . . . . . . . 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 2611 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑎 = 𝑎)
6155, 59, 60breq123d 4597 . . . . . . . . 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 480 . . . . . . . 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 246 . . . . . . 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 788 . . . . . . . 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 3569 . . . . . . 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 792 . . . . . . . 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 2611 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅s 𝐴))) → 𝑏 = 𝑏)
6855, 59, 67breq123d 4597 . . . . . . . . 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 480 . . . . . . . 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 246 . . . . . . 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 2610 . . . . . . . 8 (.r𝑅) = (.r𝑅)
7229, 57, 50, 71orngmul 29134 . . . . . . 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 1327 . . . . . 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 480 . . . . . . 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 480 . . . . . . 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 480 . . . . . . . . 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 15829 . . . . . . . . 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 6566 . . . . . . 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 4597 . . . . . 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 221 . . . . 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 449 . . . 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 677 . . 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 2954 . 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 2610 . . 3 (0g‘(𝑅s 𝐴)) = (0g‘(𝑅s 𝐴))
86 eqid 2610 . . 3 (.r‘(𝑅s 𝐴)) = (.r‘(𝑅s 𝐴))
87 eqid 2610 . . 3 (le‘(𝑅s 𝐴)) = (le‘(𝑅s 𝐴))
8820, 85, 86, 87isorng 29130 . 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 1239 1 ((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oRing)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  .rcmulr 15769  lecple 15775  0gc0g 15923  Mndcmnd 17117  Grpcgrp 17245  SubGrpcsubg 17411  1rcur 18324  Ringcrg 18370  oMndcomnd 29028  oGrpcogrp 29029  oRingcorng 29126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-ple 15788  df-0g 15925  df-poset 16769  df-toset 16857  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-subg 17414  df-mgp 18313  df-ur 18325  df-ring 18372  df-omnd 29030  df-ogrp 29031  df-orng 29128 This theorem is referenced by:  subofld  29147
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