Theorem i1fmul 25731

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 11240	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
2	1	adantl 481	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
3		i1fadd.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
4		i1ff 25711	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
6		i1fadd.2	. . . 4 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
7		i1ff 25711	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 11246	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4227	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 7715	. 2 ⊢ (𝜑 → (𝐹 ∘f · 𝐺):ℝ⟶ℝ)
13		i1frn 25712	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 25712	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 9358	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
18	14, 16, 17	syl2anc 584	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19		eqid 2737	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
20		ovex 7464	. . . . . 6 ⊢ (𝑢 · 𝑣) ∈ V
21	19, 20	fnmpoi 8095	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6826	. . . . 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 230	. . . 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 9351	. . . 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 586	. . 3 ⊢ (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
26		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) = (𝑥 · 𝑦)
27		rspceov 7480	. . . . . . . . 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 1452	. . . . . . . 8 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
29		ovex 7464	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) ∈ V
30		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 · 𝑦) → (𝑤 = (𝑢 · 𝑣) ↔ (𝑥 · 𝑦) = (𝑢 · 𝑣)))
31	30	2rexbidv 3222	. . . . . . . . 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 3679	. . . . . . . 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 234	. . . . . . 7 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
34	33	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
35	5	ffnd 6737	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
36		dffn3 6748	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
37	35, 36	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
38	8	ffnd 6737	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
39		dffn3 6748	. . . . . . 7 ⊢ (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
40	38, 39	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐺:ℝ⟶ran 𝐺)
41	34, 37, 40, 10, 10, 11	off 7715	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
42	41	frnd 6744	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
43	19	rnmpo 7566	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)}
44	42, 43	sseqtrrdi 4025	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
45	25, 44	ssfid 9301	. 2 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ∈ Fin)
46	12	frnd 6744	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ℝ)
47		ax-resscn 11212	. . . . . . 7 ⊢ ℝ ⊆ ℂ
48	46, 47	sstrdi 3996	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ℂ)
49	48	ssdifd 4145	. . . . 5 ⊢ (𝜑 → (ran (𝐹 ∘f · 𝐺) ∖ {0}) ⊆ (ℂ ∖ {0}))
50	49	sselda 3983	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
51	3, 6	i1fmullem 25729	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
52	50, 51	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
53		difss 4136	. . . . . 6 ⊢ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺
54		ssfi 9213	. . . . . 6 ⊢ ((ran 𝐺 ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺) → (ran 𝐺 ∖ {0}) ∈ Fin)
55	16, 53, 54	sylancl 586	. . . . 5 ⊢ (𝜑 → (ran 𝐺 ∖ {0}) ∈ Fin)
56		i1fima 25713	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
57	3, 56	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
58		i1fima 25713	. . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
59	6, 58	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
60		inmbl 25577	. . . . . . 7 ⊢ (((◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
61	57, 59, 60	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
62	61	ralrimivw 3150	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
63		finiunmbl 25579	. . . . 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 584	. . . 4 ⊢ (𝜑 → ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
65	64	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
66	52, 65	eqeltrd 2841	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) ∈ dom vol)
67		mblvol 25565	. . . 4 ⊢ ((◡(𝐹 ∘f · 𝐺) “ {𝑦}) ∈ dom vol → (vol‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})))
68	66, 67	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})))
69		mblss 25566	. . . . 5 ⊢ ((◡(𝐹 ∘f · 𝐺) “ {𝑦}) ∈ dom vol → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) ⊆ ℝ)
70	66, 69	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) ⊆ ℝ)
71	55	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (ran 𝐺 ∖ {0}) ∈ Fin)
72		inss2 4238	. . . . . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ∈ dom vol)
75		mblss 25566	. . . . . . 7 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (◡𝐺 “ {𝑧}) ⊆ ℝ)
76	74, 75	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
77		mblvol 25565	. . . . . . . 8 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
78	74, 77	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
79	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
80		i1fima2sn 25715	. . . . . . . 8 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
81	79, 80	sylan 580	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
82	78, 81	eqeltrrd 2842	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
83		ovolsscl 25521	. . . . . 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 1373	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
85	71, 84	fsumrecl 15770	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
86	52	fveq2d 6910	. . . . 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 25566	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
88	61, 87	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
89	88	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
90	89, 84	jca 511	. . . . . . 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 3146	. . . . . 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 25536	. . . . . 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 584	. . . . 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 5165	. . . 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 25518	. . . 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 1373	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ∈ ℝ)
97	68, 96	eqeltrd 2841	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ∈ ℝ)
98	12, 45, 66, 97	i1fd 25716	1 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ dom ∫1)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  {csn 4626  ∪ ciun 4991   class class class wbr 5143   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688   Fn wfn 6556  ⟶wf 6557  –onto→wfo 6559  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433   ∘_f cof 7695  Fincfn 8985  ℂcc 11153  ℝcr 11154  0cc0 11155   · cmul 11160   ≤ cle 11296   / cdiv 11920  Σcsu 15722  vol*covol 25497  volcvol 25498  ∫₁citg1 25650

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xadd 13155  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-xmet 21357  df-met 21358  df-ovol 25499  df-vol 25500  df-mbf 25654  df-itg1 25655

This theorem is referenced by:  mbfmullem2  25759  ftc1anclem3  37702