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Theorem suborng 31555
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 ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (𝑅 β†Ύs 𝐴) ∈ oRing)

Proof of Theorem suborng
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 486 . 2 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (𝑅 β†Ύs 𝐴) ∈ Ring)
2 ringgrp 19829 . . . 4 ((𝑅 β†Ύs 𝐴) ∈ Ring β†’ (𝑅 β†Ύs 𝐴) ∈ Grp)
32adantl 483 . . 3 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (𝑅 β†Ύs 𝐴) ∈ Grp)
4 orngogrp 31541 . . . . 5 (𝑅 ∈ oRing β†’ 𝑅 ∈ oGrp)
5 isogrp 31369 . . . . . 6 (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
65simprbi 498 . . . . 5 (𝑅 ∈ oGrp β†’ 𝑅 ∈ oMnd)
74, 6syl 17 . . . 4 (𝑅 ∈ oRing β†’ 𝑅 ∈ oMnd)
8 ringmnd 19834 . . . 4 ((𝑅 β†Ύs 𝐴) ∈ Ring β†’ (𝑅 β†Ύs 𝐴) ∈ Mnd)
9 submomnd 31377 . . . 4 ((𝑅 ∈ oMnd ∧ (𝑅 β†Ύs 𝐴) ∈ Mnd) β†’ (𝑅 β†Ύs 𝐴) ∈ oMnd)
107, 8, 9syl2an 597 . . 3 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (𝑅 β†Ύs 𝐴) ∈ oMnd)
11 isogrp 31369 . . 3 ((𝑅 β†Ύs 𝐴) ∈ oGrp ↔ ((𝑅 β†Ύs 𝐴) ∈ Grp ∧ (𝑅 β†Ύs 𝐴) ∈ oMnd))
123, 10, 11sylanbrc 584 . 2 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (𝑅 β†Ύs 𝐴) ∈ oGrp)
13 simp-4l 781 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ 𝑅 ∈ oRing)
14 reldmress 16984 . . . . . . . . . . . . . . 15 Rel dom β†Ύs
1514ovprc2 7343 . . . . . . . . . . . . . 14 (Β¬ 𝐴 ∈ V β†’ (𝑅 β†Ύs 𝐴) = βˆ…)
1615fveq2d 6804 . . . . . . . . . . . . 13 (Β¬ 𝐴 ∈ V β†’ (Baseβ€˜(𝑅 β†Ύs 𝐴)) = (Baseβ€˜βˆ…))
1716adantl 483 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ Β¬ 𝐴 ∈ V) β†’ (Baseβ€˜(𝑅 β†Ύs 𝐴)) = (Baseβ€˜βˆ…))
18 base0 16958 . . . . . . . . . . . 12 βˆ… = (Baseβ€˜βˆ…)
1917, 18eqtr4di 2794 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ Β¬ 𝐴 ∈ V) β†’ (Baseβ€˜(𝑅 β†Ύs 𝐴)) = βˆ…)
20 eqid 2736 . . . . . . . . . . . . . . 15 (Baseβ€˜(𝑅 β†Ύs 𝐴)) = (Baseβ€˜(𝑅 β†Ύs 𝐴))
21 eqid 2736 . . . . . . . . . . . . . . 15 (1rβ€˜(𝑅 β†Ύs 𝐴)) = (1rβ€˜(𝑅 β†Ύs 𝐴))
2220, 21ringidcl 19848 . . . . . . . . . . . . . 14 ((𝑅 β†Ύs 𝐴) ∈ Ring β†’ (1rβ€˜(𝑅 β†Ύs 𝐴)) ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴)))
2322ne0d 4275 . . . . . . . . . . . . 13 ((𝑅 β†Ύs 𝐴) ∈ Ring β†’ (Baseβ€˜(𝑅 β†Ύs 𝐴)) β‰  βˆ…)
2423ad2antlr 725 . . . . . . . . . . . 12 (((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ Β¬ 𝐴 ∈ V) β†’ (Baseβ€˜(𝑅 β†Ύs 𝐴)) β‰  βˆ…)
2524neneqd 2946 . . . . . . . . . . 11 (((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ Β¬ 𝐴 ∈ V) β†’ Β¬ (Baseβ€˜(𝑅 β†Ύs 𝐴)) = βˆ…)
2619, 25condan 816 . . . . . . . . . 10 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ 𝐴 ∈ V)
27 eqid 2736 . . . . . . . . . . . 12 (𝑅 β†Ύs 𝐴) = (𝑅 β†Ύs 𝐴)
28 eqid 2736 . . . . . . . . . . . 12 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2927, 28ressbas 16988 . . . . . . . . . . 11 (𝐴 ∈ V β†’ (𝐴 ∩ (Baseβ€˜π‘…)) = (Baseβ€˜(𝑅 β†Ύs 𝐴)))
30 inss2 4169 . . . . . . . . . . 11 (𝐴 ∩ (Baseβ€˜π‘…)) βŠ† (Baseβ€˜π‘…)
3129, 30eqsstrrdi 3981 . . . . . . . . . 10 (𝐴 ∈ V β†’ (Baseβ€˜(𝑅 β†Ύs 𝐴)) βŠ† (Baseβ€˜π‘…))
3226, 31syl 17 . . . . . . . . 9 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (Baseβ€˜(𝑅 β†Ύs 𝐴)) βŠ† (Baseβ€˜π‘…))
3332ad3antrrr 728 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (Baseβ€˜(𝑅 β†Ύs 𝐴)) βŠ† (Baseβ€˜π‘…))
34 simpllr 774 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴)))
3533, 34sseldd 3927 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ π‘Ž ∈ (Baseβ€˜π‘…))
36 simprl 769 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž)
37 orngring 31540 . . . . . . . . . . . . . . . 16 (𝑅 ∈ oRing β†’ 𝑅 ∈ Ring)
38 ringgrp 19829 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring β†’ 𝑅 ∈ Grp)
3937, 38syl 17 . . . . . . . . . . . . . . 15 (𝑅 ∈ oRing β†’ 𝑅 ∈ Grp)
4039adantr 482 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ 𝑅 ∈ Grp)
4128ressinbas 16996 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V β†’ (𝑅 β†Ύs 𝐴) = (𝑅 β†Ύs (𝐴 ∩ (Baseβ€˜π‘…))))
4229oveq2d 7319 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V β†’ (𝑅 β†Ύs (𝐴 ∩ (Baseβ€˜π‘…))) = (𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴))))
4341, 42eqtrd 2776 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V β†’ (𝑅 β†Ύs 𝐴) = (𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴))))
4426, 43syl 17 . . . . . . . . . . . . . . 15 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (𝑅 β†Ύs 𝐴) = (𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴))))
4544, 3eqeltrrd 2838 . . . . . . . . . . . . . 14 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∈ Grp)
4628issubg 18796 . . . . . . . . . . . . . 14 ((Baseβ€˜(𝑅 β†Ύs 𝐴)) ∈ (SubGrpβ€˜π‘…) ↔ (𝑅 ∈ Grp ∧ (Baseβ€˜(𝑅 β†Ύs 𝐴)) βŠ† (Baseβ€˜π‘…) ∧ (𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∈ Grp))
4740, 32, 45, 46syl3anbrc 1343 . . . . . . . . . . . . 13 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (Baseβ€˜(𝑅 β†Ύs 𝐴)) ∈ (SubGrpβ€˜π‘…))
48 eqid 2736 . . . . . . . . . . . . . 14 (𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴))) = (𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴)))
49 eqid 2736 . . . . . . . . . . . . . 14 (0gβ€˜π‘…) = (0gβ€˜π‘…)
5048, 49subg0 18802 . . . . . . . . . . . . 13 ((Baseβ€˜(𝑅 β†Ύs 𝐴)) ∈ (SubGrpβ€˜π‘…) β†’ (0gβ€˜π‘…) = (0gβ€˜(𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴)))))
5147, 50syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (0gβ€˜π‘…) = (0gβ€˜(𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴)))))
5244fveq2d 6804 . . . . . . . . . . . 12 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (0gβ€˜(𝑅 β†Ύs 𝐴)) = (0gβ€˜(𝑅 β†Ύs (Baseβ€˜(𝑅 β†Ύs 𝐴)))))
5351, 52eqtr4d 2779 . . . . . . . . . . 11 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (0gβ€˜π‘…) = (0gβ€˜(𝑅 β†Ύs 𝐴)))
5453ad2antrr 724 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) β†’ (0gβ€˜π‘…) = (0gβ€˜(𝑅 β†Ύs 𝐴)))
5526ad2antrr 724 . . . . . . . . . . 11 ((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) β†’ 𝐴 ∈ V)
56 eqid 2736 . . . . . . . . . . . 12 (leβ€˜π‘…) = (leβ€˜π‘…)
5727, 56ressle 17131 . . . . . . . . . . 11 (𝐴 ∈ V β†’ (leβ€˜π‘…) = (leβ€˜(𝑅 β†Ύs 𝐴)))
5855, 57syl 17 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) β†’ (leβ€˜π‘…) = (leβ€˜(𝑅 β†Ύs 𝐴)))
59 eqidd 2737 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) β†’ π‘Ž = π‘Ž)
6054, 58, 59breq123d 5095 . . . . . . . . 9 ((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) β†’ ((0gβ€˜π‘…)(leβ€˜π‘…)π‘Ž ↔ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž))
6160adantr 482 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ ((0gβ€˜π‘…)(leβ€˜π‘…)π‘Ž ↔ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž))
6236, 61mpbird 258 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (0gβ€˜π‘…)(leβ€˜π‘…)π‘Ž)
63 simplr 767 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴)))
6433, 63sseldd 3927 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ 𝑏 ∈ (Baseβ€˜π‘…))
65 simprr 771 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)
66 eqidd 2737 . . . . . . . . . 10 ((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) β†’ 𝑏 = 𝑏)
6754, 58, 66breq123d 5095 . . . . . . . . 9 ((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) β†’ ((0gβ€˜π‘…)(leβ€˜π‘…)𝑏 ↔ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏))
6867adantr 482 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ ((0gβ€˜π‘…)(leβ€˜π‘…)𝑏 ↔ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏))
6965, 68mpbird 258 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (0gβ€˜π‘…)(leβ€˜π‘…)𝑏)
70 eqid 2736 . . . . . . . 8 (.rβ€˜π‘…) = (.rβ€˜π‘…)
7128, 56, 49, 70orngmul 31543 . . . . . . 7 ((𝑅 ∈ oRing ∧ (π‘Ž ∈ (Baseβ€˜π‘…) ∧ (0gβ€˜π‘…)(leβ€˜π‘…)π‘Ž) ∧ (𝑏 ∈ (Baseβ€˜π‘…) ∧ (0gβ€˜π‘…)(leβ€˜π‘…)𝑏)) β†’ (0gβ€˜π‘…)(leβ€˜π‘…)(π‘Ž(.rβ€˜π‘…)𝑏))
7213, 35, 62, 64, 69, 71syl122anc 1379 . . . . . 6 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (0gβ€˜π‘…)(leβ€˜π‘…)(π‘Ž(.rβ€˜π‘…)𝑏))
7354adantr 482 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (0gβ€˜π‘…) = (0gβ€˜(𝑅 β†Ύs 𝐴)))
7458adantr 482 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (leβ€˜π‘…) = (leβ€˜(𝑅 β†Ύs 𝐴)))
7555adantr 482 . . . . . . . . 9 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ 𝐴 ∈ V)
7627, 70ressmulr 17058 . . . . . . . . 9 (𝐴 ∈ V β†’ (.rβ€˜π‘…) = (.rβ€˜(𝑅 β†Ύs 𝐴)))
7775, 76syl 17 . . . . . . . 8 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (.rβ€˜π‘…) = (.rβ€˜(𝑅 β†Ύs 𝐴)))
7877oveqd 7320 . . . . . . 7 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (π‘Ž(.rβ€˜π‘…)𝑏) = (π‘Ž(.rβ€˜(𝑅 β†Ύs 𝐴))𝑏))
7973, 74, 78breq123d 5095 . . . . . 6 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ ((0gβ€˜π‘…)(leβ€˜π‘…)(π‘Ž(.rβ€˜π‘…)𝑏) ↔ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))(π‘Ž(.rβ€˜(𝑅 β†Ύs 𝐴))𝑏)))
8072, 79mpbid 232 . . . . 5 (((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ ((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏)) β†’ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))(π‘Ž(.rβ€˜(𝑅 β†Ύs 𝐴))𝑏))
8180ex 414 . . . 4 ((((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))) β†’ (((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏) β†’ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))(π‘Ž(.rβ€˜(𝑅 β†Ύs 𝐴))𝑏)))
8281anasss 468 . . 3 (((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴)) ∧ 𝑏 ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴)))) β†’ (((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏) β†’ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))(π‘Ž(.rβ€˜(𝑅 β†Ύs 𝐴))𝑏)))
8382ralrimivva 3194 . 2 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ βˆ€π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))βˆ€π‘ ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))(((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏) β†’ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))(π‘Ž(.rβ€˜(𝑅 β†Ύs 𝐴))𝑏)))
84 eqid 2736 . . 3 (0gβ€˜(𝑅 β†Ύs 𝐴)) = (0gβ€˜(𝑅 β†Ύs 𝐴))
85 eqid 2736 . . 3 (.rβ€˜(𝑅 β†Ύs 𝐴)) = (.rβ€˜(𝑅 β†Ύs 𝐴))
86 eqid 2736 . . 3 (leβ€˜(𝑅 β†Ύs 𝐴)) = (leβ€˜(𝑅 β†Ύs 𝐴))
8720, 84, 85, 86isorng 31539 . 2 ((𝑅 β†Ύs 𝐴) ∈ oRing ↔ ((𝑅 β†Ύs 𝐴) ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ oGrp ∧ βˆ€π‘Ž ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))βˆ€π‘ ∈ (Baseβ€˜(𝑅 β†Ύs 𝐴))(((0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))π‘Ž ∧ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))𝑏) β†’ (0gβ€˜(𝑅 β†Ύs 𝐴))(leβ€˜(𝑅 β†Ύs 𝐴))(π‘Ž(.rβ€˜(𝑅 β†Ύs 𝐴))𝑏))))
881, 12, 83, 87syl3anbrc 1343 1 ((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (𝑅 β†Ύs 𝐴) ∈ oRing)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104   β‰  wne 2941  βˆ€wral 3062  Vcvv 3437   ∩ cin 3891   βŠ† wss 3892  βˆ…c0 4262   class class class wbr 5081  β€˜cfv 6454  (class class class)co 7303  Basecbs 16953   β†Ύs cress 16982  .rcmulr 17004  lecple 17010  0gc0g 17191  Mndcmnd 18426  Grpcgrp 18618  SubGrpcsubg 18790  1rcur 19778  Ringcrg 19824  oMndcomnd 31364  oGrpcogrp 31365  oRingcorng 31535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616  ax-cnex 10969  ax-resscn 10970  ax-1cn 10971  ax-icn 10972  ax-addcl 10973  ax-addrcl 10974  ax-mulcl 10975  ax-mulrcl 10976  ax-mulcom 10977  ax-addass 10978  ax-mulass 10979  ax-distr 10980  ax-i2m1 10981  ax-1ne0 10982  ax-1rid 10983  ax-rnegex 10984  ax-rrecex 10985  ax-cnre 10986  ax-pre-lttri 10987  ax-pre-lttrn 10988  ax-pre-ltadd 10989  ax-pre-mulgt0 10990
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5496  df-eprel 5502  df-po 5510  df-so 5511  df-fr 5551  df-we 5553  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-pred 6213  df-ord 6280  df-on 6281  df-lim 6282  df-suc 6283  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462  df-riota 7260  df-ov 7306  df-oprab 7307  df-mpo 7308  df-om 7741  df-2nd 7860  df-frecs 8124  df-wrecs 8155  df-recs 8229  df-rdg 8268  df-er 8525  df-en 8761  df-dom 8762  df-sdom 8763  df-pnf 11053  df-mnf 11054  df-xr 11055  df-ltxr 11056  df-le 11057  df-sub 11249  df-neg 11250  df-nn 12016  df-2 12078  df-3 12079  df-4 12080  df-5 12081  df-6 12082  df-7 12083  df-8 12084  df-9 12085  df-dec 12480  df-sets 16906  df-slot 16924  df-ndx 16936  df-base 16954  df-ress 16983  df-plusg 17016  df-mulr 17017  df-ple 17023  df-0g 17193  df-poset 18072  df-toset 18176  df-mgm 18367  df-sgrp 18416  df-mnd 18427  df-grp 18621  df-subg 18793  df-mgp 19762  df-ur 19779  df-ring 19826  df-omnd 31366  df-ogrp 31367  df-orng 31537
This theorem is referenced by:  subofld  31556
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