Theorem imasvscafn 16804

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 2821	. . . . . . . 8 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))
2		fvex 6677	. . . . . . . 8 ⊢ (𝐹‘(𝑝 · 𝑞)) ∈ V
3	1, 2	fnmpoi 7762	. . . . . . 7 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) Fn (𝐾 × {(𝐹‘𝑞)})
4		fnrel 6448	. . . . . . 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 3150	. . . . 5 ⊢ ∀𝑞 ∈ 𝑉 Rel (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))
7		reliun 5683	. . . . 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 233	. . . 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 16787	. . . . 5 ⊢ (𝜑 → ∙ = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
18	17	releqd 5647	. . . 4 ⊢ (𝜑 → (Rel ∙ ↔ Rel ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))))
19	8, 18	mpbiri 260	. . 3 ⊢ (𝜑 → Rel ∙ )
20		dffn2 6510	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))):(𝐾 × {(𝐹‘𝑞)})⟶V
22		fssxp 6528	. . . . . . . . . . . 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 6584	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
25	11, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
26	25	ffvelrnda 6845	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑉) → (𝐹‘𝑞) ∈ 𝐵)
27	26	snssd 4735	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑉) → {(𝐹‘𝑞)} ⊆ 𝐵)
28		xpss2 5569	. . . . . . . . . . . 12 ⊢ ({(𝐹‘𝑞)} ⊆ 𝐵 → (𝐾 × {(𝐹‘𝑞)}) ⊆ (𝐾 × 𝐵))
29		xpss1 5568	. . . . . . . . . . . 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 3977	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑉) → (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × 𝐵) × V))
32	31	ralrimiva 3182	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × 𝐵) × V))
33		iunss 4961	. . . . . . . . 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 236	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ((𝐾 × 𝐵) × V))
35	17, 34	eqsstrd 4004	. . . . . . 7 ⊢ (𝜑 → ∙ ⊆ ((𝐾 × 𝐵) × V))
36		dmss 5765	. . . . . . 7 ⊢ ( ∙ ⊆ ((𝐾 × 𝐵) × V) → dom ∙ ⊆ dom ((𝐾 × 𝐵) × V))
37	35, 36	syl 17	. . . . . 6 ⊢ (𝜑 → dom ∙ ⊆ dom ((𝐾 × 𝐵) × V))
38		vn0 4303	. . . . . . 7 ⊢ V ≠ ∅
39		dmxp 5793	. . . . . . 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 4016	. . . . 5 ⊢ (𝜑 → dom ∙ ⊆ (𝐾 × 𝐵))
42		forn 6587	. . . . . . 7 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
43	11, 42	syl 17	. . . . . 6 ⊢ (𝜑 → ran 𝐹 = 𝐵)
44	43	xpeq2d 5579	. . . . 5 ⊢ (𝜑 → (𝐾 × ran 𝐹) = (𝐾 × 𝐵))
45	41, 44	sseqtrrd 4007	. . . 4 ⊢ (𝜑 → dom ∙ ⊆ (𝐾 × ran 𝐹))
46		df-br 5059	. . . . . . . . . 10 ⊢ (⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤 ↔ ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∙ )
47	17	eleq2d 2898	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∙ ↔ ⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))))
48	47	adantr 483	. . . . . . . . . . 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 4915	. . . . . . . . . . . 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 1085	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉) ↔ ((𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉))
51	1	mpofun 7270	. . . . . . . . . . . . . . . . . . . 20 ⊢ Fun (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))
52		funopfv 6711	. . . . . . . . . . . . . . . . . . . 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 7153	. . . . . . . . . . . . . . . . . . . 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 5348	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ⟨𝑝, (𝐹‘𝑎)⟩ ∈ V
56		vex 3497	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑤 ∈ V
57	55, 56	opeldm 5770	. . . . . . . . . . . . . . . . . . . . . . 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 7763	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ dom (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = (𝐾 × {(𝐹‘𝑞)})
59	57, 58	eleqtrdi 2923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → ⟨𝑝, (𝐹‘𝑎)⟩ ∈ (𝐾 × {(𝐹‘𝑞)}))
60		opelxp 5585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑝, (𝐹‘𝑎)⟩ ∈ (𝐾 × {(𝐹‘𝑞)}) ↔ (𝑝 ∈ 𝐾 ∧ (𝐹‘𝑎) ∈ {(𝐹‘𝑞)}))
61	59, 60	sylib 220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → (𝑝 ∈ 𝐾 ∧ (𝐹‘𝑎) ∈ {(𝐹‘𝑞)}))
62		fvoveq1 7173	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑝 → (𝐹‘(𝑧 · 𝑞)) = (𝐹‘(𝑝 · 𝑞)))
63		eqidd 2822	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝐹‘𝑎) → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑞)))
64		fvoveq1 7173	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = 𝑧 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑧 · 𝑞)))
65		eqidd 2822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑧 · 𝑞)) = (𝐹‘(𝑧 · 𝑞)))
66	64, 65	cbvmpov 7243	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = (𝑧 ∈ 𝐾, 𝑦 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑧 · 𝑞)))
67	62, 63, 66, 2	ovmpo 7304	. . . . . . . . . . . . . . . . . . . . 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	syl5eqr 2870	. . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑝 · 𝑞)))
71	70	adantl 484	. . . . . . . . . . . . . . . . 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 4577	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑎) ∈ {(𝐹‘𝑞)} → (𝐹‘𝑎) = (𝐹‘𝑞))
74	61, 73	simpl2im 506	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) → (𝐹‘𝑎) = (𝐹‘𝑞))
75	72, 74	impel 508	. . . . . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 470	. . . . . . . . . . . . 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 3284	. . . . . . . . . . . 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	syl5bi 244	. . . . . . . . . . 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 242	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → (⟨⟨𝑝, (𝐹‘𝑎)⟩, 𝑤⟩ ∈ ∙ → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
83	46, 82	syl5bi 244	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → (⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤 → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
84	83	alrimiv 1924	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉)) → ∀𝑤(⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤 → 𝑤 = (𝐹‘(𝑝 · 𝑎))))
85		mo2icl 3704	. . . . . . . 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 3191	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ 𝐾 ∀𝑎 ∈ 𝑉 ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤)
88		fofn 6586	. . . . . . . 8 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
89		opeq2 4797	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑎) → ⟨𝑝, 𝑦⟩ = ⟨𝑝, (𝐹‘𝑎)⟩)
90	89	breq1d 5068	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑎) → (⟨𝑝, 𝑦⟩ ∙ 𝑤 ↔ ⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤))
91	90	mobidv 2629	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑎) → (∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤 ↔ ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤))
92	91	ralrn 6848	. . . . . . . 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 3197	. . . . . 6 ⊢ (𝜑 → (∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤 ↔ ∀𝑝 ∈ 𝐾 ∀𝑎 ∈ 𝑉 ∃*𝑤⟨𝑝, (𝐹‘𝑎)⟩ ∙ 𝑤))
95	87, 94	mpbird 259	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤)
96		breq1 5061	. . . . . . 7 ⊢ (𝑥 = ⟨𝑝, 𝑦⟩ → (𝑥 ∙ 𝑤 ↔ ⟨𝑝, 𝑦⟩ ∙ 𝑤))
97	96	mobidv 2629	. . . . . 6 ⊢ (𝑥 = ⟨𝑝, 𝑦⟩ → (∃*𝑤 𝑥 ∙ 𝑤 ↔ ∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤))
98	97	ralxp 5706	. . . . 5 ⊢ (∀𝑥 ∈ (𝐾 × ran 𝐹)∃*𝑤 𝑥 ∙ 𝑤 ↔ ∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹∃*𝑤⟨𝑝, 𝑦⟩ ∙ 𝑤)
99	95, 98	sylibr 236	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐾 × ran 𝐹)∃*𝑤 𝑥 ∙ 𝑤)
100		ssralv 4032	. . . 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 6376	. . 3 ⊢ (Fun ∙ ↔ (Rel ∙ ∧ ∀𝑥 ∈ dom ∙ ∃*𝑤 𝑥 ∙ 𝑤))
103	19, 101, 102	sylanbrc 585	. 2 ⊢ (𝜑 → Fun ∙ )
104		eqimss2 4023	. . . . . . . . . . . . . . 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 4961	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ )
108	107	r19.21bi 3208	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑉) → (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ )
109	108	adantrl 714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → (𝑝 ∈ 𝐾, 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ⊆ ∙ )
110		dmss 5765	. . . . . . . . . . 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 4015	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → (𝐾 × {(𝐹‘𝑞)}) ⊆ dom ∙ )
113		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → 𝑝 ∈ 𝐾)
114		fvex 6677	. . . . . . . . . . 11 ⊢ (𝐹‘𝑞) ∈ V
115	114	snid 4594	. . . . . . . . . 10 ⊢ (𝐹‘𝑞) ∈ {(𝐹‘𝑞)}
116		opelxpi 5586	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝐾 ∧ (𝐹‘𝑞) ∈ {(𝐹‘𝑞)}) → ⟨𝑝, (𝐹‘𝑞)⟩ ∈ (𝐾 × {(𝐹‘𝑞)}))
117	113, 115, 116	sylancl 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → ⟨𝑝, (𝐹‘𝑞)⟩ ∈ (𝐾 × {(𝐹‘𝑞)}))
118	112, 117	sseldd 3967	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉)) → ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ )
119	118	ralrimivva 3191	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ 𝐾 ∀𝑞 ∈ 𝑉 ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ )
120		opeq2 4797	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑞) → ⟨𝑝, 𝑦⟩ = ⟨𝑝, (𝐹‘𝑞)⟩)
121	120	eleq1d 2897	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑞) → (⟨𝑝, 𝑦⟩ ∈ dom ∙ ↔ ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ ))
122	121	ralrn 6848	. . . . . . . . 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 3197	. . . . . . 7 ⊢ (𝜑 → (∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹⟨𝑝, 𝑦⟩ ∈ dom ∙ ↔ ∀𝑝 ∈ 𝐾 ∀𝑞 ∈ 𝑉 ⟨𝑝, (𝐹‘𝑞)⟩ ∈ dom ∙ ))
125	119, 124	mpbird 259	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹⟨𝑝, 𝑦⟩ ∈ dom ∙ )
126		eleq1 2900	. . . . . . 7 ⊢ (𝑥 = ⟨𝑝, 𝑦⟩ → (𝑥 ∈ dom ∙ ↔ ⟨𝑝, 𝑦⟩ ∈ dom ∙ ))
127	126	ralxp 5706	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐾 × ran 𝐹)𝑥 ∈ dom ∙ ↔ ∀𝑝 ∈ 𝐾 ∀𝑦 ∈ ran 𝐹⟨𝑝, 𝑦⟩ ∈ dom ∙ )
128	125, 127	sylibr 236	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐾 × ran 𝐹)𝑥 ∈ dom ∙ )
129		dfss3 3955	. . . . 5 ⊢ ((𝐾 × ran 𝐹) ⊆ dom ∙ ↔ ∀𝑥 ∈ (𝐾 × ran 𝐹)𝑥 ∈ dom ∙ )
130	128, 129	sylibr 236	. . . 4 ⊢ (𝜑 → (𝐾 × ran 𝐹) ⊆ dom ∙ )
131	44, 130	eqsstrrd 4005	. . 3 ⊢ (𝜑 → (𝐾 × 𝐵) ⊆ dom ∙ )
132	41, 131	eqssd 3983	. 2 ⊢ (𝜑 → dom ∙ = (𝐾 × 𝐵))
133		df-fn 6352	. 2 ⊢ ( ∙ Fn (𝐾 × 𝐵) ↔ (Fun ∙ ∧ dom ∙ = (𝐾 × 𝐵)))
134	103, 132, 133	sylanbrc 585	1 ⊢ (𝜑 → ∙ Fn (𝐾 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1531   = wceq 1533   ∈ wcel 2110  ∃*wmo 2616   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ⊆ wss 3935  ∅c0 4290  {csn 4560  ⟨cop 4566  ∪ ciun 4911   class class class wbr 5058   × cxp 5547  dom cdm 5549  ran crn 5550  Rel wrel 5554  Fun wfun 6343   Fn wfn 6344  ⟶wf 6345  –onto→wfo 6347  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  Basecbs 16477  Scalarcsca 16562   ·_𝑠 cvsca 16563   “_s cimas 16771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-imas 16775

This theorem is referenced by:  imasvscaval  16805  imasvscaf  16806