Theorem i1fmul 25924

Description: The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)

Hypotheses

Ref	Expression
i1fadd.1	⊢ (𝜑 → 𝐹 ∈ dom ∫1)
i1fadd.2	⊢ (𝜑 → 𝐺 ∈ dom ∫1)

Assertion

Ref	Expression
i1fmul	⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ dom ∫1)

Proof of Theorem i1fmul

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

Step	Hyp	Ref	Expression
1		remulcl 11209	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
2	1	adantl 487	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
3		i1fadd.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
4		i1ff 25904	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
5	3, 4	syl 18	. . 3 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
6		i1fadd.2	. . . 4 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
7		i1ff 25904	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 18	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 11215	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4172	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 7696	. 2 ⊢ (𝜑 → (𝐹 ∘f · 𝐺):ℝ⟶ℝ)
13		i1frn 25905	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 25905	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 9289	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
18	14, 16, 17	syl2anc 596	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19		eqid 2760	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
20		ovex 7446	. . . . . 6 ⊢ (𝑢 · 𝑣) ∈ V
21	19, 20	fnmpoi 8067	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6795	. . . . 5 ⊢ ((𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺) ↔ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)):(ran 𝐹 × ran 𝐺)–onto→ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
23	21, 22	mpbi 233	. . . 4 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)):(ran 𝐹 × ran 𝐺)–onto→ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
24		fofi 9283	. . . 4 ⊢ (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)):(ran 𝐹 × ran 𝐺)–onto→ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))) → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
25	18, 23, 24	sylancl 598	. . 3 ⊢ (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
26		eqid 2760	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) = (𝑥 · 𝑦)
27		rspceov 7462	. . . . . . . . 9 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺 ∧ (𝑥 · 𝑦) = (𝑥 · 𝑦)) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
28	26, 27	mp3an3 1479	. . . . . . . 8 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
29		ovex 7446	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) ∈ V
30		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 · 𝑦) → (𝑤 = (𝑢 · 𝑣) ↔ (𝑥 · 𝑦) = (𝑢 · 𝑣)))
31	30	2rexbidv 3227	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 · 𝑦) → (∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣) ↔ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣)))
32	29, 31	elab 3633	. . . . . . . 8 ⊢ ((𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)} ↔ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
33	28, 32	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
34	33	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
35	5	ffnd 6703	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
36		dffn3 6715	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
37	35, 36	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
38	8	ffnd 6703	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
39		dffn3 6715	. . . . . . 7 ⊢ (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
40	38, 39	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐺:ℝ⟶ran 𝐺)
41	34, 37, 40, 10, 10, 11	off 7696	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
42	41	frnd 6711	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
43	19	rnmpo 7546	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)}
44	42, 43	sseqtrrdi 3972	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
45	25, 44	ssfid 9239	. 2 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ∈ Fin)
46	12	frnd 6711	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ℝ)
47		ax-resscn 11181	. . . . . . 7 ⊢ ℝ ⊆ ℂ
48	46, 47	sstrdi 3943	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ℂ)
49	48	ssdifd 4092	. . . . 5 ⊢ (𝜑 → (ran (𝐹 ∘f · 𝐺) ∖ {0}) ⊆ (ℂ ∖ {0}))
50	49	sselda 3931	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
51	3, 6	i1fmullem 25922	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
52	50, 51	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
53		difss 4083	. . . . . 6 ⊢ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺
54		ssfi 9167	. . . . . 6 ⊢ ((ran 𝐺 ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺) → (ran 𝐺 ∖ {0}) ∈ Fin)
55	16, 53, 54	sylancl 598	. . . . 5 ⊢ (𝜑 → (ran 𝐺 ∖ {0}) ∈ Fin)
56		i1fima 25906	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
57	3, 56	syl 18	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
58		i1fima 25906	. . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
59	6, 58	syl 18	. . . . . . 7 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
60		inmbl 25770	. . . . . . 7 ⊢ (((◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
61	57, 59, 60	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
62	61	ralrimivw 3158	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
63		finiunmbl 25772	. . . . 5 ⊢ (((ran 𝐺 ∖ {0}) ∈ Fin ∧ ∀𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol) → ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
64	55, 62, 63	syl2anc 596	. . . 4 ⊢ (𝜑 → ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
65	64	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
66	52, 65	eqeltrd 2860	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) ∈ dom vol)
67		mblvol 25758	. . . 4 ⊢ ((◡(𝐹 ∘f · 𝐺) “ {𝑦}) ∈ dom vol → (vol‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})))
68	66, 67	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})))
69		mblss 25759	. . . . 5 ⊢ ((◡(𝐹 ∘f · 𝐺) “ {𝑦}) ∈ dom vol → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) ⊆ ℝ)
70	66, 69	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) ⊆ ℝ)
71	55	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (ran 𝐺 ∖ {0}) ∈ Fin)
72		inss2 4183	. . . . . . 7 ⊢ ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧})
73	72	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧}))
74	59	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ∈ dom vol)
75		mblss 25759	. . . . . . 7 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (◡𝐺 “ {𝑧}) ⊆ ℝ)
76	74, 75	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
77		mblvol 25758	. . . . . . . 8 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
78	74, 77	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
79	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
80		i1fima2sn 25908	. . . . . . . 8 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
81	79, 80	sylan 592	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
82	78, 81	eqeltrrd 2861	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
83		ovolsscl 25714	. . . . . 6 ⊢ ((((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧}) ∧ (◡𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
84	73, 76, 82, 83	syl3anc 1398	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
85	71, 84	fsumrecl 15820	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
86	52	fveq2d 6882	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) = (vol*‘∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
87		mblss 25759	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
88	61, 87	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
89	88	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
90	89, 84	jca 521	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
91	90	ralrimiva 3154	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → ∀𝑧 ∈ (ran 𝐺 ∖ {0})(((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
92		ovolfiniun 25729	. . . . . 6 ⊢ (((ran 𝐺 ∖ {0}) ∈ Fin ∧ ∀𝑧 ∈ (ran 𝐺 ∖ {0})(((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)) → (vol*‘∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
93	71, 91, 92	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol*‘∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
94	86, 93	eqbrtrd 5127	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
95		ovollecl 25711	. . . 4 ⊢ (((◡(𝐹 ∘f · 𝐺) “ {𝑦}) ⊆ ℝ ∧ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ ∧ (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})))) → (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ∈ ℝ)
96	70, 85, 94, 95	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ∈ ℝ)
97	68, 96	eqeltrd 2860	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ∈ ℝ)
98	12, 45, 66, 97	i1fd 25909	1 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ dom ∫1)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  {csn 4584  ∪ ciun 4951   class class class wbr 5103   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658   Fn wfn 6528  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415   ∘_f cof 7676  Fincfn 8952  ℂcc 11122  ℝcr 11123  0cc0 11124   · cmul 11129   ≤ cle 11268   / cdiv 11895  Σcsu 15773  vol*covol 25690  volcvol 25691  ∫₁citg1 25843

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-pm 8829  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-oi 9482  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-z 12616  df-uz 12888  df-q 12998  df-rp 13043  df-xadd 13164  df-ioo 13402  df-ico 13404  df-icc 13405  df-fz 13562  df-fzo 13710  df-fl 13853  df-seq 14066  df-exp 14126  df-hash 14395  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-sum 15774  df-xmet 21578  df-met 21579  df-ovol 25692  df-vol 25693  df-mbf 25847  df-itg1 25848

This theorem is used by:  mbfmullem2  25952  ftc1anclem3  38444