Metamath Proof Explorer


Theorem issubrng

Description: The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025)

Ref Expression
Hypothesis issubrng.b B = Base R
Assertion issubrng A SubRng R R Rng R 𝑠 A Rng A B

Proof

Step Hyp Ref Expression
1 issubrng.b B = Base R
2 df-subrng SubRng = w Rng s 𝒫 Base w | w 𝑠 s Rng
3 2 mptrcl A SubRng R R Rng
4 simp1 R Rng R 𝑠 A Rng A B R Rng
5 fveq2 r = R Base r = Base R
6 5 pweqd r = R 𝒫 Base r = 𝒫 Base R
7 oveq1 r = R r 𝑠 s = R 𝑠 s
8 7 eleq1d r = R r 𝑠 s Rng R 𝑠 s Rng
9 6 8 rabeqbidv r = R s 𝒫 Base r | r 𝑠 s Rng = s 𝒫 Base R | R 𝑠 s Rng
10 df-subrng SubRng = r Rng s 𝒫 Base r | r 𝑠 s Rng
11 fvex Base R V
12 11 pwex 𝒫 Base R V
13 12 rabex s 𝒫 Base R | R 𝑠 s Rng V
14 9 10 13 fvmpt R Rng SubRng R = s 𝒫 Base R | R 𝑠 s Rng
15 14 eleq2d R Rng A SubRng R A s 𝒫 Base R | R 𝑠 s Rng
16 oveq2 s = A R 𝑠 s = R 𝑠 A
17 16 eleq1d s = A R 𝑠 s Rng R 𝑠 A Rng
18 17 elrab A s 𝒫 Base R | R 𝑠 s Rng A 𝒫 Base R R 𝑠 A Rng
19 1 eqcomi Base R = B
20 19 sseq2i A Base R A B
21 20 anbi2i R 𝑠 A Rng A Base R R 𝑠 A Rng A B
22 ibar R Rng R 𝑠 A Rng A B R Rng R 𝑠 A Rng A B
23 21 22 bitrid R Rng R 𝑠 A Rng A Base R R Rng R 𝑠 A Rng A B
24 11 elpw2 A 𝒫 Base R A Base R
25 24 anbi2ci A 𝒫 Base R R 𝑠 A Rng R 𝑠 A Rng A Base R
26 3anass R Rng R 𝑠 A Rng A B R Rng R 𝑠 A Rng A B
27 23 25 26 3bitr4g R Rng A 𝒫 Base R R 𝑠 A Rng R Rng R 𝑠 A Rng A B
28 18 27 bitrid R Rng A s 𝒫 Base R | R 𝑠 s Rng R Rng R 𝑠 A Rng A B
29 15 28 bitrd R Rng A SubRng R R Rng R 𝑠 A Rng A B
30 3 4 29 pm5.21nii A SubRng R R Rng R 𝑠 A Rng A B