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Theorem issrg 13153
Description: The predicate "is a semiring". (Contributed by Thierry Arnoux, 21-Mar-2018.)
Hypotheses
Ref Expression
issrg.b 𝐡 = (Baseβ€˜π‘…)
issrg.g 𝐺 = (mulGrpβ€˜π‘…)
issrg.p + = (+gβ€˜π‘…)
issrg.t Β· = (.rβ€˜π‘…)
issrg.0 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
issrg (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))))
Distinct variable groups:   π‘₯,𝑦,𝑧, +   π‘₯, 0 ,𝑦,𝑧   π‘₯, Β· ,𝑦,𝑧   π‘₯,𝐡,𝑦,𝑧   π‘₯,𝑅,𝑦,𝑧
Allowed substitution hints:   𝐺(π‘₯,𝑦,𝑧)
Proof of Theorem issrg
Dummy variables 𝑛 𝑏 𝑝 π‘Ÿ 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2750 . 2 (𝑅 ∈ SRing β†’ 𝑅 ∈ V)
2 simp1 997 . . 3 ((𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))) β†’ 𝑅 ∈ CMnd)
32elexd 2752 . 2 ((𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))) β†’ 𝑅 ∈ V)
4 issrg.g . . . . . . . 8 𝐺 = (mulGrpβ€˜π‘…)
54eleq1i 2243 . . . . . . 7 (𝐺 ∈ Mnd ↔ (mulGrpβ€˜π‘…) ∈ Mnd)
65bicomi 132 . . . . . 6 ((mulGrpβ€˜π‘…) ∈ Mnd ↔ 𝐺 ∈ Mnd)
76a1i 9 . . . . 5 (𝑅 ∈ V β†’ ((mulGrpβ€˜π‘…) ∈ Mnd ↔ 𝐺 ∈ Mnd))
8 issrg.b . . . . . . 7 𝐡 = (Baseβ€˜π‘…)
9 basfn 12522 . . . . . . . 8 Base Fn V
10 funfvex 5534 . . . . . . . . 9 ((Fun Base ∧ 𝑅 ∈ dom Base) β†’ (Baseβ€˜π‘…) ∈ V)
1110funfni 5318 . . . . . . . 8 ((Base Fn V ∧ 𝑅 ∈ V) β†’ (Baseβ€˜π‘…) ∈ V)
129, 11mpan 424 . . . . . . 7 (𝑅 ∈ V β†’ (Baseβ€˜π‘…) ∈ V)
138, 12eqeltrid 2264 . . . . . 6 (𝑅 ∈ V β†’ 𝐡 ∈ V)
14 issrg.p . . . . . . . . 9 + = (+gβ€˜π‘…)
15 plusgslid 12573 . . . . . . . . . 10 (+g = Slot (+gβ€˜ndx) ∧ (+gβ€˜ndx) ∈ β„•)
1615slotex 12491 . . . . . . . . 9 (𝑅 ∈ V β†’ (+gβ€˜π‘…) ∈ V)
1714, 16eqeltrid 2264 . . . . . . . 8 (𝑅 ∈ V β†’ + ∈ V)
1817adantr 276 . . . . . . 7 ((𝑅 ∈ V ∧ 𝑏 = 𝐡) β†’ + ∈ V)
19 issrg.t . . . . . . . . . 10 Β· = (.rβ€˜π‘…)
20 mulrslid 12592 . . . . . . . . . . 11 (.r = Slot (.rβ€˜ndx) ∧ (.rβ€˜ndx) ∈ β„•)
2120slotex 12491 . . . . . . . . . 10 (𝑅 ∈ V β†’ (.rβ€˜π‘…) ∈ V)
2219, 21eqeltrid 2264 . . . . . . . . 9 (𝑅 ∈ V β†’ Β· ∈ V)
2322ad2antrr 488 . . . . . . . 8 (((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) β†’ Β· ∈ V)
24 issrg.0 . . . . . . . . . . 11 0 = (0gβ€˜π‘…)
25 fn0g 12799 . . . . . . . . . . . 12 0g Fn V
26 funfvex 5534 . . . . . . . . . . . . 13 ((Fun 0g ∧ 𝑅 ∈ dom 0g) β†’ (0gβ€˜π‘…) ∈ V)
2726funfni 5318 . . . . . . . . . . . 12 ((0g Fn V ∧ 𝑅 ∈ V) β†’ (0gβ€˜π‘…) ∈ V)
2825, 27mpan 424 . . . . . . . . . . 11 (𝑅 ∈ V β†’ (0gβ€˜π‘…) ∈ V)
2924, 28eqeltrid 2264 . . . . . . . . . 10 (𝑅 ∈ V β†’ 0 ∈ V)
3029ad3antrrr 492 . . . . . . . . 9 ((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) β†’ 0 ∈ V)
31 simp-4r 542 . . . . . . . . . 10 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ 𝑏 = 𝐡)
32 simplr 528 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ 𝑑 = Β· )
33 eqidd 2178 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ π‘₯ = π‘₯)
34 simpllr 534 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ 𝑝 = + )
3534oveqd 5894 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (𝑦𝑝𝑧) = (𝑦 + 𝑧))
3632, 33, 35oveq123d 5898 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (π‘₯𝑑(𝑦𝑝𝑧)) = (π‘₯ Β· (𝑦 + 𝑧)))
3732oveqd 5894 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (π‘₯𝑑𝑦) = (π‘₯ Β· 𝑦))
3832oveqd 5894 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (π‘₯𝑑𝑧) = (π‘₯ Β· 𝑧))
3934, 37, 38oveq123d 5898 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))
4036, 39eqeq12d 2192 . . . . . . . . . . . . . 14 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ↔ (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧))))
4134oveqd 5894 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (π‘₯𝑝𝑦) = (π‘₯ + 𝑦))
42 eqidd 2178 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ 𝑧 = 𝑧)
4332, 41, 42oveq123d 5898 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯ + 𝑦) Β· 𝑧))
4432oveqd 5894 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (𝑦𝑑𝑧) = (𝑦 Β· 𝑧))
4534, 38, 44oveq123d 5898 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧)) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))
4643, 45eqeq12d 2192 . . . . . . . . . . . . . 14 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧)) ↔ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))))
4740, 46anbi12d 473 . . . . . . . . . . . . 13 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ↔ ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))))
4831, 47raleqbidv 2685 . . . . . . . . . . . 12 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ↔ βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))))
4931, 48raleqbidv 2685 . . . . . . . . . . 11 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ↔ βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))))
50 simpr 110 . . . . . . . . . . . . . 14 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ 𝑛 = 0 )
5132, 50, 33oveq123d 5898 . . . . . . . . . . . . 13 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (𝑛𝑑π‘₯) = ( 0 Β· π‘₯))
5251, 50eqeq12d 2192 . . . . . . . . . . . 12 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ ((𝑛𝑑π‘₯) = 𝑛 ↔ ( 0 Β· π‘₯) = 0 ))
5332, 33, 50oveq123d 5898 . . . . . . . . . . . . 13 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (π‘₯𝑑𝑛) = (π‘₯ Β· 0 ))
5453, 50eqeq12d 2192 . . . . . . . . . . . 12 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ ((π‘₯𝑑𝑛) = 𝑛 ↔ (π‘₯ Β· 0 ) = 0 ))
5552, 54anbi12d 473 . . . . . . . . . . 11 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛) ↔ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 )))
5649, 55anbi12d 473 . . . . . . . . . 10 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ ((βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))))
5731, 56raleqbidv 2685 . . . . . . . . 9 (((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) ∧ 𝑛 = 0 ) β†’ (βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))))
5830, 57sbcied 3001 . . . . . . . 8 ((((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) ∧ 𝑑 = Β· ) β†’ ([ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))))
5923, 58sbcied 3001 . . . . . . 7 (((𝑅 ∈ V ∧ 𝑏 = 𝐡) ∧ 𝑝 = + ) β†’ ([ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))))
6018, 59sbcied 3001 . . . . . 6 ((𝑅 ∈ V ∧ 𝑏 = 𝐡) β†’ ([ + / 𝑝][ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))))
6113, 60sbcied 3001 . . . . 5 (𝑅 ∈ V β†’ ([𝐡 / 𝑏][ + / 𝑝][ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))))
627, 61anbi12d 473 . . . 4 (𝑅 ∈ V β†’ (((mulGrpβ€˜π‘…) ∈ Mnd ∧ [𝐡 / 𝑏][ + / 𝑝][ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛))) ↔ (𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 )))))
6362anbi2d 464 . . 3 (𝑅 ∈ V β†’ ((𝑅 ∈ CMnd ∧ ((mulGrpβ€˜π‘…) ∈ Mnd ∧ [𝐡 / 𝑏][ + / 𝑝][ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)))) ↔ (𝑅 ∈ CMnd ∧ (𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))))))
64 fveq2 5517 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
6564eleq1d 2246 . . . . 5 (π‘Ÿ = 𝑅 β†’ ((mulGrpβ€˜π‘Ÿ) ∈ Mnd ↔ (mulGrpβ€˜π‘…) ∈ Mnd))
66 fveq2 5517 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
6766, 8eqtr4di 2228 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
68 fveq2 5517 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (+gβ€˜π‘Ÿ) = (+gβ€˜π‘…))
6968, 14eqtr4di 2228 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (+gβ€˜π‘Ÿ) = + )
70 fveq2 5517 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
7170, 19eqtr4di 2228 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = Β· )
72 fveq2 5517 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = (0gβ€˜π‘…))
7372, 24eqtr4di 2228 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = 0 )
7473sbceq1d 2969 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ ([(0gβ€˜π‘Ÿ) / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ [ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛))))
7571, 74sbceqbid 2971 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ([(.rβ€˜π‘Ÿ) / 𝑑][(0gβ€˜π‘Ÿ) / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ [ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛))))
7669, 75sbceqbid 2971 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ([(+gβ€˜π‘Ÿ) / 𝑝][(.rβ€˜π‘Ÿ) / 𝑑][(0gβ€˜π‘Ÿ) / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ [ + / 𝑝][ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛))))
7767, 76sbceqbid 2971 . . . . 5 (π‘Ÿ = 𝑅 β†’ ([(Baseβ€˜π‘Ÿ) / 𝑏][(+gβ€˜π‘Ÿ) / 𝑝][(.rβ€˜π‘Ÿ) / 𝑑][(0gβ€˜π‘Ÿ) / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)) ↔ [𝐡 / 𝑏][ + / 𝑝][ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛))))
7865, 77anbi12d 473 . . . 4 (π‘Ÿ = 𝑅 β†’ (((mulGrpβ€˜π‘Ÿ) ∈ Mnd ∧ [(Baseβ€˜π‘Ÿ) / 𝑏][(+gβ€˜π‘Ÿ) / 𝑝][(.rβ€˜π‘Ÿ) / 𝑑][(0gβ€˜π‘Ÿ) / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛))) ↔ ((mulGrpβ€˜π‘…) ∈ Mnd ∧ [𝐡 / 𝑏][ + / 𝑝][ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)))))
79 df-srg 13152 . . . 4 SRing = {π‘Ÿ ∈ CMnd ∣ ((mulGrpβ€˜π‘Ÿ) ∈ Mnd ∧ [(Baseβ€˜π‘Ÿ) / 𝑏][(+gβ€˜π‘Ÿ) / 𝑝][(.rβ€˜π‘Ÿ) / 𝑑][(0gβ€˜π‘Ÿ) / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)))}
8078, 79elrab2 2898 . . 3 (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ ((mulGrpβ€˜π‘…) ∈ Mnd ∧ [𝐡 / 𝑏][ + / 𝑝][ Β· / 𝑑][ 0 / 𝑛]βˆ€π‘₯ ∈ 𝑏 (βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 ((π‘₯𝑑(𝑦𝑝𝑧)) = ((π‘₯𝑑𝑦)𝑝(π‘₯𝑑𝑧)) ∧ ((π‘₯𝑝𝑦)𝑑𝑧) = ((π‘₯𝑑𝑧)𝑝(𝑦𝑑𝑧))) ∧ ((𝑛𝑑π‘₯) = 𝑛 ∧ (π‘₯𝑑𝑛) = 𝑛)))))
81 3anass 982 . . 3 ((𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))) ↔ (𝑅 ∈ CMnd ∧ (𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 )))))
8263, 80, 813bitr4g 223 . 2 (𝑅 ∈ V β†’ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 )))))
831, 3, 82pm5.21nii 704 1 (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ βˆ€π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)) ∧ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧))) ∧ (( 0 Β· π‘₯) = 0 ∧ (π‘₯ Β· 0 ) = 0 ))))
Colors of variables: wff set class
Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  Vcvv 2739  [wsbc 2964   Fn wfn 5213  β€˜cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  .rcmulr 12539  0gc0g 12710  Mndcmnd 12822  CMndccmn 13093  mulGrpcmgp 13135  SRingcsrg 13151
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
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 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-mulr 12552  df-0g 12712  df-srg 13152
This theorem is referenced by:  srgcmn  13154  srgmgp  13156  srgdilem  13157  srgrz  13172  srglz  13173  ringsrg  13229
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