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Theorem issrg 12961
Description: The predicate "is a semiring". (Contributed by Thierry Arnoux, 21-Mar-2018.)
Hypotheses
Ref Expression
issrg.b 𝐵 = (Base‘𝑅)
issrg.g 𝐺 = (mulGrp‘𝑅)
issrg.p + = (+g𝑅)
issrg.t · = (.r𝑅)
issrg.0 0 = (0g𝑅)
Assertion
Ref Expression
issrg (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
Distinct variable groups:   𝑥,𝑦,𝑧, +   𝑥, 0 ,𝑦,𝑧   𝑥, · ,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem issrg
Dummy variables 𝑛 𝑏 𝑝 𝑟 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2748 . 2 (𝑅 ∈ SRing → 𝑅 ∈ V)
2 simp1 997 . . 3 ((𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))) → 𝑅 ∈ CMnd)
32elexd 2750 . 2 ((𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))) → 𝑅 ∈ V)
4 issrg.g . . . . . . . 8 𝐺 = (mulGrp‘𝑅)
54eleq1i 2243 . . . . . . 7 (𝐺 ∈ Mnd ↔ (mulGrp‘𝑅) ∈ Mnd)
65bicomi 132 . . . . . 6 ((mulGrp‘𝑅) ∈ Mnd ↔ 𝐺 ∈ Mnd)
76a1i 9 . . . . 5 (𝑅 ∈ V → ((mulGrp‘𝑅) ∈ Mnd ↔ 𝐺 ∈ Mnd))
8 issrg.b . . . . . . 7 𝐵 = (Base‘𝑅)
9 basfn 12489 . . . . . . . 8 Base Fn V
10 funfvex 5527 . . . . . . . . 9 ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
1110funfni 5311 . . . . . . . 8 ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
129, 11mpan 424 . . . . . . 7 (𝑅 ∈ V → (Base‘𝑅) ∈ V)
138, 12eqeltrid 2264 . . . . . 6 (𝑅 ∈ V → 𝐵 ∈ V)
14 issrg.p . . . . . . . . 9 + = (+g𝑅)
15 plusgslid 12538 . . . . . . . . . 10 (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
1615slotex 12459 . . . . . . . . 9 (𝑅 ∈ V → (+g𝑅) ∈ V)
1714, 16eqeltrid 2264 . . . . . . . 8 (𝑅 ∈ V → + ∈ V)
1817adantr 276 . . . . . . 7 ((𝑅 ∈ V ∧ 𝑏 = 𝐵) → + ∈ V)
19 issrg.t . . . . . . . . . 10 · = (.r𝑅)
20 mulrslid 12556 . . . . . . . . . . 11 (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
2120slotex 12459 . . . . . . . . . 10 (𝑅 ∈ V → (.r𝑅) ∈ V)
2219, 21eqeltrid 2264 . . . . . . . . 9 (𝑅 ∈ V → · ∈ V)
2322ad2antrr 488 . . . . . . . 8 (((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → · ∈ V)
24 issrg.0 . . . . . . . . . . 11 0 = (0g𝑅)
25 fn0g 12673 . . . . . . . . . . . 12 0g Fn V
26 funfvex 5527 . . . . . . . . . . . . 13 ((Fun 0g𝑅 ∈ dom 0g) → (0g𝑅) ∈ V)
2726funfni 5311 . . . . . . . . . . . 12 ((0g Fn V ∧ 𝑅 ∈ V) → (0g𝑅) ∈ V)
2825, 27mpan 424 . . . . . . . . . . 11 (𝑅 ∈ V → (0g𝑅) ∈ V)
2924, 28eqeltrid 2264 . . . . . . . . . 10 (𝑅 ∈ V → 0 ∈ V)
3029ad3antrrr 492 . . . . . . . . 9 ((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 0 ∈ V)
31 simp-4r 542 . . . . . . . . . 10 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑏 = 𝐵)
32 simplr 528 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑡 = · )
33 eqidd 2178 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑥 = 𝑥)
34 simpllr 534 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑝 = + )
3534oveqd 5885 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑦𝑝𝑧) = (𝑦 + 𝑧))
3632, 33, 35oveq123d 5889 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡(𝑦𝑝𝑧)) = (𝑥 · (𝑦 + 𝑧)))
3732oveqd 5885 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡𝑦) = (𝑥 · 𝑦))
3832oveqd 5885 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡𝑧) = (𝑥 · 𝑧))
3934, 37, 38oveq123d 5889 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
4036, 39eqeq12d 2192 . . . . . . . . . . . . . 14 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ↔ (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))))
4134oveqd 5885 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
42 eqidd 2178 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑧 = 𝑧)
4332, 41, 42oveq123d 5889 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥 + 𝑦) · 𝑧))
4432oveqd 5885 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑦𝑡𝑧) = (𝑦 · 𝑧))
4534, 38, 44oveq123d 5889 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
4643, 45eqeq12d 2192 . . . . . . . . . . . . . 14 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) ↔ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
4740, 46anbi12d 473 . . . . . . . . . . . . 13 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
4831, 47raleqbidv 2684 . . . . . . . . . . . 12 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (∀𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
4931, 48raleqbidv 2684 . . . . . . . . . . 11 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
50 simpr 110 . . . . . . . . . . . . . 14 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑛 = 0 )
5132, 50, 33oveq123d 5889 . . . . . . . . . . . . 13 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑛𝑡𝑥) = ( 0 · 𝑥))
5251, 50eqeq12d 2192 . . . . . . . . . . . 12 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑛𝑡𝑥) = 𝑛 ↔ ( 0 · 𝑥) = 0 ))
5332, 33, 50oveq123d 5889 . . . . . . . . . . . . 13 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡𝑛) = (𝑥 · 0 ))
5453, 50eqeq12d 2192 . . . . . . . . . . . 12 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡𝑛) = 𝑛 ↔ (𝑥 · 0 ) = 0 ))
5552, 54anbi12d 473 . . . . . . . . . . 11 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛) ↔ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))
5649, 55anbi12d 473 . . . . . . . . . 10 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
5731, 56raleqbidv 2684 . . . . . . . . 9 (((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (∀𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
5830, 57sbcied 2999 . . . . . . . 8 ((((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ([ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
5923, 58sbcied 2999 . . . . . . 7 (((𝑅 ∈ V ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → ([ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
6018, 59sbcied 2999 . . . . . 6 ((𝑅 ∈ V ∧ 𝑏 = 𝐵) → ([ + / 𝑝][ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
6113, 60sbcied 2999 . . . . 5 (𝑅 ∈ V → ([𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
627, 61anbi12d 473 . . . 4 (𝑅 ∈ V → (((mulGrp‘𝑅) ∈ Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))) ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))))
6362anbi2d 464 . . 3 (𝑅 ∈ V → ((𝑅 ∈ CMnd ∧ ((mulGrp‘𝑅) ∈ Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))) ↔ (𝑅 ∈ CMnd ∧ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))))
64 fveq2 5510 . . . . . 6 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
6564eleq1d 2246 . . . . 5 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ Mnd ↔ (mulGrp‘𝑅) ∈ Mnd))
66 fveq2 5510 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
6766, 8eqtr4di 2228 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
68 fveq2 5510 . . . . . . . 8 (𝑟 = 𝑅 → (+g𝑟) = (+g𝑅))
6968, 14eqtr4di 2228 . . . . . . 7 (𝑟 = 𝑅 → (+g𝑟) = + )
70 fveq2 5510 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
7170, 19eqtr4di 2228 . . . . . . . 8 (𝑟 = 𝑅 → (.r𝑟) = · )
72 fveq2 5510 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
7372, 24eqtr4di 2228 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = 0 )
7473sbceq1d 2967 . . . . . . . 8 (𝑟 = 𝑅 → ([(0g𝑟) / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))
7571, 74sbceqbid 2969 . . . . . . 7 (𝑟 = 𝑅 → ([(.r𝑟) / 𝑡][(0g𝑟) / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))
7669, 75sbceqbid 2969 . . . . . 6 (𝑟 = 𝑅 → ([(+g𝑟) / 𝑝][(.r𝑟) / 𝑡][(0g𝑟) / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [ + / 𝑝][ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))
7767, 76sbceqbid 2969 . . . . 5 (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑏][(+g𝑟) / 𝑝][(.r𝑟) / 𝑡][(0g𝑟) / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))
7865, 77anbi12d 473 . . . 4 (𝑟 = 𝑅 → (((mulGrp‘𝑟) ∈ Mnd ∧ [(Base‘𝑟) / 𝑏][(+g𝑟) / 𝑝][(.r𝑟) / 𝑡][(0g𝑟) / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))) ↔ ((mulGrp‘𝑅) ∈ Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))))
79 df-srg 12960 . . . 4 SRing = {𝑟 ∈ CMnd ∣ ((mulGrp‘𝑟) ∈ Mnd ∧ [(Base‘𝑟) / 𝑏][(+g𝑟) / 𝑝][(.r𝑟) / 𝑡][(0g𝑟) / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))}
8078, 79elrab2 2896 . . 3 (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ ((mulGrp‘𝑅) ∈ Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]𝑥𝑏 (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))))
81 3anass 982 . . 3 ((𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))) ↔ (𝑅 ∈ CMnd ∧ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))))
8263, 80, 813bitr4g 223 . 2 (𝑅 ∈ V → (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))))
831, 3, 82pm5.21nii 704 1 (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wral 2455  Vcvv 2737  [wsbc 2962   Fn wfn 5206  cfv 5211  (class class class)co 5868  Basecbs 12432  +gcplusg 12505  .rcmulr 12506  0gc0g 12640  Mndcmnd 12696  CMndccmn 12902  mulGrpcmgp 12944  SRingcsrg 12959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-mulr 12519  df-0g 12642  df-srg 12960
This theorem is referenced by:  srgcmn  12962  srgmgp  12964  srgdilem  12965  srgrz  12980  srglz  12981  ringsrg  13037
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