Metamath Proof Explorer

Theorem psrgrpOLD

Description: Obsolete version of psrgrp as of 7-Feb-2025. (Contributed by Mario Carneiro, 29-Dec-2014) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses psrgrp.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
psrgrp.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
psrgrp.r ⊢ ( 𝜑 → 𝑅 ∈ Grp )
Assertion psrgrpOLD ( 𝜑 → 𝑆 ∈ Grp )

Proof

Step Hyp Ref Expression
1 psrgrp.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psrgrp.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
3 psrgrp.r ⊢ ( 𝜑 → 𝑅 ∈ Grp )
4 eqidd ⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) )
5 eqidd ⊢ ( 𝜑 → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) )
6 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
7 eqid ⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
8 3 grpmgmd ⊢ ( 𝜑 → 𝑅 ∈ Mgm )
9 8 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑅 ∈ Mgm )
10 simp2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
11 simp3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
12 1 6 7 9 10 11 psraddcl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
13 ovex ⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
14 13 rabex ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V
15 14 a1i ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V )
16 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
17 eqid ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
18 simpr1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
19 1 16 17 6 18 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑥 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
20 simpr2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
21 1 16 17 6 20 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑦 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
22 simpr3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 ∈ ( Base ‘ 𝑆 ) )
23 1 16 17 6 22 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑧 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
24 3 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑅 ∈ Grp )
25 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
26 16 25 grpass ⊢ ( ( 𝑅 ∈ Grp ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑟 ( +g ‘ 𝑅 ) 𝑠 ) ( +g ‘ 𝑅 ) 𝑡 ) = ( 𝑟 ( +g ‘ 𝑅 ) ( 𝑠 ( +g ‘ 𝑅 ) 𝑡 ) ) )
27 24 26 sylan ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) ∧ ( 𝑟 ∈ ( Base ‘ 𝑅 ) ∧ 𝑠 ∈ ( Base ‘ 𝑅 ) ∧ 𝑡 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑟 ( +g ‘ 𝑅 ) 𝑠 ) ( +g ‘ 𝑅 ) 𝑡 ) = ( 𝑟 ( +g ‘ 𝑅 ) ( 𝑠 ( +g ‘ 𝑅 ) 𝑡 ) ) )
28 15 19 21 23 27 caofass ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ∘f ( +g ‘ 𝑅 ) 𝑧 ) ) )
29 1 6 25 7 18 20 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) )
30 29 oveq1d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) = ( ( 𝑥 ∘f ( +g ‘ 𝑅 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) )
31 1 6 25 7 20 22 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) = ( 𝑦 ∘f ( +g ‘ 𝑅 ) 𝑧 ) )
32 31 oveq2d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ∘f ( +g ‘ 𝑅 ) 𝑧 ) ) )
33 28 30 32 3eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) )
34 12 3adant3r3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
35 1 6 25 7 34 22 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( +g ‘ 𝑆 ) 𝑧 ) = ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∘f ( +g ‘ 𝑅 ) 𝑧 ) )
36 8 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑅 ∈ Mgm )
37 1 6 7 36 20 22 psraddcl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ∈ ( Base ‘ 𝑆 ) )
38 1 6 25 7 18 37 psradd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) = ( 𝑥 ∘f ( +g ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) )
39 33 35 38 3eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( +g ‘ 𝑆 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝑆 ) ( 𝑦 ( +g ‘ 𝑆 ) 𝑧 ) ) )
40 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
41 1 2 3 17 40 6 psr0cl ⊢ ( 𝜑 → ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) ∈ ( Base ‘ 𝑆 ) )
42 2 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝐼 ∈ 𝑉 )
43 3 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝑅 ∈ Grp )
44 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
45 1 42 43 17 40 6 7 44 psr0lid ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) ( +g ‘ 𝑆 ) 𝑥 ) = 𝑥 )
46 eqid ⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 )
47 1 42 43 17 46 6 44 psrnegcl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( invg ‘ 𝑅 ) ∘ 𝑥 ) ∈ ( Base ‘ 𝑆 ) )
48 1 42 43 17 46 6 44 40 7 psrlinv ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( invg ‘ 𝑅 ) ∘ 𝑥 ) ( +g ‘ 𝑆 ) 𝑥 ) = ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) )
49 4 5 12 39 41 45 47 48 isgrpd ⊢ ( 𝜑 → 𝑆 ∈ Grp )