Theorem psrvscafval

Description: The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014) (Revised by Mario Carneiro, 2-Oct-2015) (Proof shortened by AV, 2-Nov-2024)

	Ref	Expression
Hypotheses	psrvsca.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
	psrvsca.n	⊢ ∙ = ( ·𝑠 ‘ 𝑆 )
	psrvsca.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	psrvsca.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	psrvsca.m	⊢ · = ( .r ‘ 𝑅 )
	psrvsca.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
Assertion	psrvscafval	⊢ ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) )

Proof

Step	Hyp	Ref	Expression
1		psrvsca.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2		psrvsca.n	⊢ ∙ = ( ·𝑠 ‘ 𝑆 )
3		psrvsca.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
4		psrvsca.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
5		psrvsca.m	⊢ · = ( .r ‘ 𝑅 )
6		psrvsca.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
7		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
8		eqid	⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 𝑅 )
9		simpl	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐼 ∈ V )
10	1 3 6 4 9	psrbas	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ( 𝐾 ↑m 𝐷 ) )
11		eqid	⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
12	1 4 7 11	psrplusg	⊢ ( +g ‘ 𝑆 ) = ( ∘f ( +g ‘ 𝑅 ) ↾ ( 𝐵 × 𝐵 ) )
13		eqid	⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
14	1 4 5 13 6	psrmulr	⊢ ( .r ‘ 𝑆 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )
15		eqid	⊢ ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) )
16		eqidd	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) = ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) )
17		simpr	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑅 ∈ V )
18	1 3 7 5 8 6 10 12 14 15 16 9 17	psrval	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑆 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) )
19	18	fveq2d	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) ) )
20	3	fvexi	⊢ 𝐾 ∈ V
21	4	fvexi	⊢ 𝐵 ∈ V
22	20 21	mpoex	⊢ ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ∈ V
23		psrvalstr	⊢ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) Struct ⟨ 1 , 9 ⟩
24		vscaid	⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx )
25		snsstp2	⊢ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ } ⊆ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ }
26		ssun2	⊢ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } )
27	25 26	sstri	⊢ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } )
28	23 24 27	strfv	⊢ ( ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ∈ V → ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) ) )
29	22 28	ax-mp	⊢ ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) )
30	19 2 29	3eqtr4g	⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) )
31		eqid	⊢ ∅ = ∅
32		fn0	⊢ ( ∅ Fn ∅ ↔ ∅ = ∅ )
33	31 32	mpbir	⊢ ∅ Fn ∅
34		reldmpsr	⊢ Rel dom mPwSer
35	34	ovprc	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mPwSer 𝑅 ) = ∅ )
36	1 35	eqtrid	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑆 = ∅ )
37	36	fveq2d	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ ∅ ) )
38	24	str0	⊢ ∅ = ( ·𝑠 ‘ ∅ )
39	37 2 38	3eqtr4g	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ∅ )
40	34 1 4	elbasov	⊢ ( 𝑓 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) )
41	40	con3i	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ¬ 𝑓 ∈ 𝐵 )
42	41	eq0rdv	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ )
43	42	xpeq2d	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐾 × 𝐵 ) = ( 𝐾 × ∅ ) )
44		xp0	⊢ ( 𝐾 × ∅ ) = ∅
45	43 44	eqtrdi	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐾 × 𝐵 ) = ∅ )
46	39 45	fneq12d	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ∙ Fn ( 𝐾 × 𝐵 ) ↔ ∅ Fn ∅ ) )
47	33 46	mpbiri	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ Fn ( 𝐾 × 𝐵 ) )
48		fnov	⊢ ( ∙ Fn ( 𝐾 × 𝐵 ) ↔ ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( 𝑥 ∙ 𝑓 ) ) )
49	47 48	sylib	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( 𝑥 ∙ 𝑓 ) ) )
50	41	pm2.21d	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑓 ∈ 𝐵 → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( 𝑥 ∙ 𝑓 ) ) )
51	50	a1d	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑥 ∈ 𝐾 → ( 𝑓 ∈ 𝐵 → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( 𝑥 ∙ 𝑓 ) ) ) )
52	51	3imp	⊢ ( ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑥 ∈ 𝐾 ∧ 𝑓 ∈ 𝐵 ) → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( 𝑥 ∙ 𝑓 ) )
53	52	mpoeq3dva	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( 𝑥 ∙ 𝑓 ) ) )
54	49 53	eqtr4d	⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) )
55	30 54	pm2.61i	⊢ ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) )