Theorem i1fmul 25597

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 11153	. . . 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 25577	. . . 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 25577	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 11159	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4190	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 7671	. 2 ⊢ (𝜑 → (𝐹 ∘f · 𝐺):ℝ⟶ℝ)
13		i1frn 25578	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 25578	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 9269	. . . . 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 2729	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
20		ovex 7420	. . . . . 6 ⊢ (𝑢 · 𝑣) ∈ V
21	19, 20	fnmpoi 8049	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6778	. . . . 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 9262	. . . 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 2729	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) = (𝑥 · 𝑦)
27		rspceov 7436	. . . . . . . . 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 7420	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) ∈ V
30		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 · 𝑦) → (𝑤 = (𝑢 · 𝑣) ↔ (𝑥 · 𝑦) = (𝑢 · 𝑣)))
31	30	2rexbidv 3202	. . . . . . . . 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 3646	. . . . . . . 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 6689	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
36		dffn3 6700	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
37	35, 36	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
38	8	ffnd 6689	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
39		dffn3 6700	. . . . . . 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 7671	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
42	41	frnd 6696	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
43	19	rnmpo 7522	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)}
44	42, 43	sseqtrrdi 3988	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
45	25, 44	ssfid 9212	. 2 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ∈ Fin)
46	12	frnd 6696	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ℝ)
47		ax-resscn 11125	. . . . . . 7 ⊢ ℝ ⊆ ℂ
48	46, 47	sstrdi 3959	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ℂ)
49	48	ssdifd 4108	. . . . 5 ⊢ (𝜑 → (ran (𝐹 ∘f · 𝐺) ∖ {0}) ⊆ (ℂ ∖ {0}))
50	49	sselda 3946	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
51	3, 6	i1fmullem 25595	. . . 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 4099	. . . . . 6 ⊢ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺
54		ssfi 9137	. . . . . 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 25579	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
57	3, 56	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
58		i1fima 25579	. . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
59	6, 58	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
60		inmbl 25443	. . . . . . 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 3129	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
63		finiunmbl 25445	. . . . 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 2828	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) ∈ dom vol)
67		mblvol 25431	. . . 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 25432	. . . . 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 4201	. . . . . . 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 25432	. . . . . . 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 25431	. . . . . . . 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 25581	. . . . . . . 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 2829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
83		ovolsscl 25387	. . . . . 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 15700	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
86	52	fveq2d 6862	. . . . 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 25432	. . . . . . . . . 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 3125	. . . . . 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 25402	. . . . . 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 5129	. . . 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 25384	. . . 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 2828	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ∈ ℝ)
98	12, 45, 66, 97	i1fd 25582	1 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ dom ∫1)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914  {csn 4589  ∪ ciun 4955   class class class wbr 5107   × cxp 5636  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641   Fn wfn 6506  ⟶wf 6507  –onto→wfo 6509  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389   ∘_f cof 7651  Fincfn 8918  ℂcc 11066  ℝcr 11067  0cc0 11068   · cmul 11073   ≤ cle 11209   / cdiv 11835  Σcsu 15652  vol*covol 25363  volcvol 25364  ∫₁citg1 25516

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-inf2 9594  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  ax-pre-sup 11146

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-rmo 3354  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-op 4596  df-uni 4872  df-int 4911  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-se 5592  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-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  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-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-oi 9463  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-q 12908  df-rp 12952  df-xadd 13073  df-ioo 13310  df-ico 13312  df-icc 13313  df-fz 13469  df-fzo 13616  df-fl 13754  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-sum 15653  df-xmet 21257  df-met 21258  df-ovol 25365  df-vol 25366  df-mbf 25520  df-itg1 25521

This theorem is referenced by:  mbfmullem2  25625  ftc1anclem3  37689