Theorem imasvscafn 17500

Description: The image structure's scalar multiplication is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
imasvscaf.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasvscaf.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasvscaf.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasvscaf.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasvscaf.g	⊢ 𝐺 = (Scalar‘𝑅)
imasvscaf.k	⊢ 𝐾 = (Base‘𝐺)
imasvscaf.q	⊢ · = ( ·𝑠 ‘𝑅)
imasvscaf.s	⊢ ∙ = ( ·𝑠 ‘𝑈)
imasvscaf.e	⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑞) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞))))

Assertion

Ref	Expression
imasvscafn	⊢ (𝜑 → ∙ Fn (𝐾 × 𝐵))

Distinct variable groups:   𝑝,𝑎,𝑞,𝐹   𝐾,𝑎,𝑝,𝑞   𝜑,𝑎,𝑝,𝑞   𝐵,𝑝,𝑞   𝑅,𝑝,𝑞   · ,𝑝,𝑞   ∙ ,𝑎,𝑝,𝑞   𝑉,𝑎,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑎)   𝑅(𝑎)   · (𝑎)   𝑈(𝑞,𝑝,𝑎)   𝐺(𝑞,𝑝,𝑎)   𝑍(𝑞,𝑝,𝑎)

Proof of Theorem imasvscafn

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . . . . . 8 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))
2		fvex 6871	. . . . . . . 8 ⊢ (𝐹‘(𝑝 · 𝑞)) ∈ V
3	1, 2	fnmpoi 8049	. . . . . . 7 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) Fn (𝐾 × {(𝐹‘𝑞)})
4		fnrel 6620	. . . . . . 7 ⊢ ((𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) Fn (𝐾 × {(𝐹‘𝑞)}) → Rel (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ Rel (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))
6	5	rgenw 3048	. . . . 5 ⊢ ∀𝑞 ∈ 𝑉 Rel (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))
7		reliun 5779	. . . . 5 ⊢ (Rel ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ↔ ∀𝑞 ∈ 𝑉 Rel (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
8	6, 7	mpbir 231	. . . 4 ⊢ Rel ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))
9		imasvscaf.u	. . . . . 6 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
10		imasvscaf.v	. . . . . 6 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
11		imasvscaf.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
12		imasvscaf.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
13		imasvscaf.g	. . . . . 6 ⊢ 𝐺 = (Scalar‘𝑅)
14		imasvscaf.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐺)
15		imasvscaf.q	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑅)
16		imasvscaf.s	. . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝑈)
17	9, 10, 11, 12, 13, 14, 15, 16	imasvsca 17483	. . . . 5 ⊢ (𝜑 → ∙ = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
18	17	releqd 5741	. . . 4 ⊢ (𝜑 → (Rel ∙ ↔ Rel ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))))
19	8, 18	mpbiri 258	. . 3 ⊢ (𝜑 → Rel ∙ )
20		dffn2 6690	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) Fn (𝐾 × {(𝐹‘𝑞)}) ↔ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))):(𝐾 × {(𝐹‘𝑞)})⟶V)
21	3, 20	mpbi 230	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))):(𝐾 × {(𝐹‘𝑞)})⟶V
22		fssxp 6715	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))):(𝐾 × {(𝐹‘𝑞)})⟶V → (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × {(𝐹‘𝑞)}) × V))
23	21, 22	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × {(𝐹‘𝑞)}) × V)
24		fof 6772	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
25	11, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
26	25	ffvelcdmda 7056	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑉) → (𝐹‘𝑞) ∈ 𝐵)
27	26	snssd 4773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑉) → {(𝐹‘𝑞)} ⊆ 𝐵)
28		xpss2 5658	. . . . . . . . . . . 12 ⊢ ({(𝐹‘𝑞)} ⊆ 𝐵 → (𝐾 × {(𝐹‘𝑞)}) ⊆ (𝐾 × 𝐵))
29		xpss1 5657	. . . . . . . . . . . 12 ⊢ ((𝐾 × {(𝐹‘𝑞)}) ⊆ (𝐾 × 𝐵) → ((𝐾 × {(𝐹‘𝑞)}) × V) ⊆ ((𝐾 × 𝐵) × V))
30	27, 28, 29	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑉) → ((𝐾 × {(𝐹‘𝑞)}) × V) ⊆ ((𝐾 × 𝐵) × V))
31	23, 30	sstrid 3958	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑉) → (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × 𝐵) × V))
32	31	ralrimiva 3125	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × 𝐵) × V))
33		iunss 5009	. . . . . . . . 9 ⊢ (∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × 𝐵) × V) ↔ ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × 𝐵) × V))
34	32, 33	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × 𝐵) × V))
35	17, 34	eqsstrd 3981	. . . . . . 7 ⊢ (𝜑 → ∙ ⊆ ((𝐾 × 𝐵) × V))
36		dmss 5866	. . . . . . 7 ⊢ ( ∙ ⊆ ((𝐾 × 𝐵) × V) → dom ∙ ⊆ dom ((𝐾 × 𝐵) × V))
37	35, 36	syl 17	. . . . . 6 ⊢ (𝜑 → dom ∙ ⊆ dom ((𝐾 × 𝐵) × V))
38		vn0 4308	. . . . . . 7 ⊢ V ≠ ∅
39		dmxp 5892	. . . . . . 7 ⊢ (V ≠ ∅ → dom ((𝐾 × 𝐵) × V) = (𝐾 × 𝐵))
40	38, 39	ax-mp 5	. . . . . 6 ⊢ dom ((𝐾 × 𝐵) × V) = (𝐾 × 𝐵)
41	37, 40	sseqtrdi 3987	. . . . 5 ⊢ (𝜑 → dom ∙ ⊆ (𝐾 × 𝐵))
42		forn 6775	. . . . . . 7 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
43	11, 42	syl 17	. . . . . 6 ⊢ (𝜑 → ran 𝐹 = 𝐵)
44	43	xpeq2d 5668	. . . . 5 ⊢ (𝜑 → (𝐾 × ran 𝐹) = (𝐾 × 𝐵))
45	41, 44	sseqtrrd 3984	. . . 4 ⊢ (𝜑 → dom ∙ ⊆ (𝐾 × ran 𝐹))
46		df-br 5108	. . . . . . . . . 10 ⊢ (⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤 ↔ ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∙ )
47	17	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∙ ↔ ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))))
48	47	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∙ ↔ ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))))
49		eliun 4959	. . . . . . . . . . . 12 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ↔ ∃𝑞 ∈ 𝑉 ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
50		df-3an 1088	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉) ↔ ((𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉))
51	1	mpofun 7513	. . . . . . . . . . . . . . . . . . . 20 ⊢ Fun (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))
52		funopfv 6910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Fun (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → ((𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))‘⟨𝑝, (𝐹‘𝑎)⟩) = 𝑤))
53	51, 52	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → ((𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))‘⟨𝑝, (𝐹‘𝑎)⟩) = 𝑤)
54		df-ov 7390	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝(𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))(𝐹‘𝑎)) = ((𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))‘⟨𝑝, (𝐹‘𝑎)⟩)
55		opex 5424	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ⟨𝑝, (𝐹‘𝑎)⟩ ∈ V
56		vex 3451	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑤 ∈ V
57	55, 56	opeldm 5871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → ⟨𝑝, (𝐹‘𝑎)⟩ ∈ dom (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
58	1, 2	dmmpo 8050	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ dom (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = (𝐾 × {(𝐹‘𝑞)})
59	57, 58	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → ⟨𝑝, (𝐹‘𝑎)⟩ ∈ (𝐾 × {(𝐹‘𝑞)}))
60		opelxp 5674	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑝, (𝐹‘𝑎)⟩ ∈ (𝐾 × {(𝐹‘𝑞)}) ↔ (𝑝 ∈ 𝐾 ∧ (𝐹‘𝑎) ∈ {(𝐹‘𝑞)}))
61	59, 60	sylib 218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → (𝑝 ∈ 𝐾 ∧ (𝐹‘𝑎) ∈ {(𝐹‘𝑞)}))
62		fvoveq1 7410	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑝 → (𝐹‘(𝑧 · 𝑞)) = (𝐹‘(𝑝 · 𝑞)))
63		eqidd 2730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝐹‘𝑎) → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑞)))
64		fvoveq1 7410	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = 𝑧 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑧 · 𝑞)))
65		eqidd 2730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑧 · 𝑞)) = (𝐹‘(𝑧 · 𝑞)))
66	64, 65	cbvmpov 7484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = (𝑧 ∈ 𝐾, 𝑦 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑧 · 𝑞)))
67	62, 63, 66, 2	ovmpo 7549	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 ∈ 𝐾 ∧ (𝐹‘𝑎) ∈ {(𝐹‘𝑞)}) → (𝑝(𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))(𝐹‘𝑎)) = (𝐹‘(𝑝 · 𝑞)))
68	61, 67	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → (𝑝(𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))(𝐹‘𝑎)) = (𝐹‘(𝑝 · 𝑞)))
69	54, 68	eqtr3id 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → ((𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))‘⟨𝑝, (𝐹‘𝑎)⟩) = (𝐹‘(𝑝 · 𝑞)))
70	53, 69	eqtr3d 2766	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑝 · 𝑞)))
71	70	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) ∧ ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))) → 𝑤 = (𝐹‘(𝑝 · 𝑞)))
72		imasvscaf.e	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑞) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞))))
73		elsni 4606	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑎) ∈ {(𝐹‘𝑞)} → (𝐹‘𝑎) = (𝐹‘𝑞))
74	61, 73	simpl2im 503	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → (𝐹‘𝑎) = (𝐹‘𝑞))
75	72, 74	impel 505	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) ∧ ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞)))
76	71, 75	eqtr4d 2767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) ∧ ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))) → 𝑤 = (𝐹‘(𝑝 · 𝑎)))
77	76	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
78	50, 77	sylan2br 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉)) → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
79	78	anassrs 467	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) ∧ 𝑞 ∈ 𝑉) → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
80	79	rexlimdva 3134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → (∃𝑞 ∈ 𝑉 ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
81	49, 80	biimtrid 242	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
82	48, 81	sylbid 240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∙ → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
83	46, 82	biimtrid 242	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → (⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤 → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
84	83	alrimiv 1927	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → ∀𝑤(⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤 → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
85		mo2icl 3685	. . . . . . . 8 ⊢ (∀𝑤(⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤 → 𝑤 = (𝐹‘(𝑝 · 𝑎))) → ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤)
86	84, 85	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤)
87	86	ralrimivva 3180	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ 𝐾 ∀𝑎 ∈ 𝑉 ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤)
88		fofn 6774	. . . . . . . 8 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
89		opeq2 4838	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑎) → ⟨𝑝, 𝑦⟩ = ⟨𝑝, (𝐹‘𝑎)⟩)
90	89	breq1d 5117	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑎) → (⟨𝑝, 𝑦⟩ ∙ 𝑤 ↔ ⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤))
91	90	mobidv 2542	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑎) → (∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤 ↔ ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤))
92	91	ralrn 7060	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤 ↔ ∀𝑎 ∈ 𝑉 ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤))
93	11, 88, 92	3syl 18	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤 ↔ ∀𝑎 ∈ 𝑉 ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤))
94	93	ralbidv 3156	. . . . . 6 ⊢ (𝜑 → (∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤 ↔ ∀𝑝 ∈ 𝐾 ∀𝑎 ∈ 𝑉 ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤))
95	87, 94	mpbird 257	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤)
96		breq1 5110	. . . . . . 7 ⊢ (𝑥 = ⟨𝑝, 𝑦⟩ → (𝑥 ∙ 𝑤 ↔ ⟨𝑝, 𝑦⟩ ∙ 𝑤))
97	96	mobidv 2542	. . . . . 6 ⊢ (𝑥 = ⟨𝑝, 𝑦⟩ → (∃*𝑤 𝑥 ∙ 𝑤 ↔ ∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤))
98	97	ralxp 5805	. . . . 5 ⊢ (∀𝑥 ∈ (𝐾 × ran 𝐹)∃*𝑤 𝑥 ∙ 𝑤 ↔ ∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤)
99	95, 98	sylibr 234	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐾 × ran 𝐹)∃*𝑤 𝑥 ∙ 𝑤)
100		ssralv 4015	. . . 4 ⊢ (dom ∙ ⊆ (𝐾 × ran 𝐹) → (∀𝑥 ∈ (𝐾 × ran 𝐹)∃*𝑤 𝑥 ∙ 𝑤 → ∀𝑥 ∈ dom ∙ ∃*𝑤 𝑥 ∙ 𝑤))
101	45, 99, 100	sylc 65	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ dom ∙ ∃*𝑤 𝑥 ∙ 𝑤)
102		dffun7 6543	. . 3 ⊢ (Fun ∙ ↔ (Rel ∙ ∧ ∀𝑥 ∈ dom ∙ ∃*𝑤 𝑥 ∙ 𝑤))
103	19, 101, 102	sylanbrc 583	. 2 ⊢ (𝜑 → Fun ∙ )
104		eqimss2 4006	. . . . . . . . . . . . . . 15 ⊢ ( ∙ = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ )
105	17, 104	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ )
106		iunss 5009	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ ↔ ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ )
107	105, 106	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ )
108	107	r19.21bi 3229	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑉) → (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ )
109	108	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ )
110		dmss 5866	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ → dom (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ dom ∙ )
111	109, 110	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → dom (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ dom ∙ )
112	58, 111	eqsstrrid 3986	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → (𝐾 × {(𝐹‘𝑞)}) ⊆ dom ∙ )
113		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → 𝑝 ∈ 𝐾)
114		fvex 6871	. . . . . . . . . . 11 ⊢ (𝐹‘𝑞) ∈ V
115	114	snid 4626	. . . . . . . . . 10 ⊢ (𝐹‘𝑞) ∈ {(𝐹‘𝑞)}
116		opelxpi 5675	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝐾 ∧ (𝐹‘𝑞) ∈ {(𝐹‘𝑞)}) → ⟨𝑝, (𝐹‘𝑞)⟩ ∈ (𝐾 × {(𝐹‘𝑞)}))
117	113, 115, 116	sylancl 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → ⟨𝑝, (𝐹‘𝑞)⟩ ∈ (𝐾 × {(𝐹‘𝑞)}))
118	112, 117	sseldd 3947	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ )
119	118	ralrimivva 3180	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ 𝐾 ∀𝑞 ∈ 𝑉 ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ )
120		opeq2 4838	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑞) → ⟨𝑝, 𝑦⟩ = ⟨𝑝, (𝐹‘𝑞)⟩)
121	120	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑞) → (⟨𝑝, 𝑦⟩ ∈ dom ∙ ↔ ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ ))
122	121	ralrn 7060	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹⟨𝑝, 𝑦⟩ ∈ dom ∙ ↔ ∀𝑞 ∈ 𝑉 ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ ))
123	11, 88, 122	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹⟨𝑝, 𝑦⟩ ∈ dom ∙ ↔ ∀𝑞 ∈ 𝑉 ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ ))
124	123	ralbidv 3156	. . . . . . 7 ⊢ (𝜑 → (∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹⟨𝑝, 𝑦⟩ ∈ dom ∙ ↔ ∀𝑝 ∈ 𝐾 ∀𝑞 ∈ 𝑉 ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ ))
125	119, 124	mpbird 257	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹⟨𝑝, 𝑦⟩ ∈ dom ∙ )
126		eleq1 2816	. . . . . . 7 ⊢ (𝑥 = ⟨𝑝, 𝑦⟩ → (𝑥 ∈ dom ∙ ↔ ⟨𝑝, 𝑦⟩ ∈ dom ∙ ))
127	126	ralxp 5805	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐾 × ran 𝐹)𝑥 ∈ dom ∙ ↔ ∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹⟨𝑝, 𝑦⟩ ∈ dom ∙ )
128	125, 127	sylibr 234	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐾 × ran 𝐹)𝑥 ∈ dom ∙ )
129		dfss3 3935	. . . . 5 ⊢ ((𝐾 × ran 𝐹) ⊆ dom ∙ ↔ ∀𝑥 ∈ (𝐾 × ran 𝐹)𝑥 ∈ dom ∙ )
130	128, 129	sylibr 234	. . . 4 ⊢ (𝜑 → (𝐾 × ran 𝐹) ⊆ dom ∙ )
131	44, 130	eqsstrrd 3982	. . 3 ⊢ (𝜑 → (𝐾 × 𝐵) ⊆ dom ∙ )
132	41, 131	eqssd 3964	. 2 ⊢ (𝜑 → dom ∙ = (𝐾 × 𝐵))
133		df-fn 6514	. 2 ⊢ ( ∙ Fn (𝐾 × 𝐵) ↔ (Fun ∙ ∧ dom ∙ = (𝐾 × 𝐵)))
134	103, 132, 133	sylanbrc 583	1 ⊢ (𝜑 → ∙ Fn (𝐾 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  {csn 4589  ⟨cop 4595  ∪ ciun 4955   class class class wbr 5107   × cxp 5636  dom cdm 5638  ran crn 5639  Rel wrel 5643  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  –onto→wfo 6509  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224   “_s cimas 17467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-imas 17471

This theorem is referenced by:  imasvscaval  17501  imasvscaf  17502