Theorem i1fmul 25750

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 11269	. . . 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 25730	. . . 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 25730	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 11275	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4248	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 7732	. 2 ⊢ (𝜑 → (𝐹 ∘f · 𝐺):ℝ⟶ℝ)
13		i1frn 25731	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 25731	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 9386	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
18	14, 16, 17	syl2anc 583	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19		eqid 2740	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
20		ovex 7481	. . . . . 6 ⊢ (𝑢 · 𝑣) ∈ V
21	19, 20	fnmpoi 8111	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6840	. . . . 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 9379	. . . 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 585	. . 3 ⊢ (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
26		eqid 2740	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) = (𝑥 · 𝑦)
27		rspceov 7497	. . . . . . . . 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 1450	. . . . . . . 8 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
29		ovex 7481	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) ∈ V
30		eqeq1 2744	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 · 𝑦) → (𝑤 = (𝑢 · 𝑣) ↔ (𝑥 · 𝑦) = (𝑢 · 𝑣)))
31	30	2rexbidv 3228	. . . . . . . . 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 3694	. . . . . . . 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 6748	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
36		dffn3 6759	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
37	35, 36	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
38	8	ffnd 6748	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
39		dffn3 6759	. . . . . . 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 7732	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
42	41	frnd 6755	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
43	19	rnmpo 7583	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)}
44	42, 43	sseqtrrdi 4060	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
45	25, 44	ssfid 9329	. 2 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ∈ Fin)
46	12	frnd 6755	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ℝ)
47		ax-resscn 11241	. . . . . . 7 ⊢ ℝ ⊆ ℂ
48	46, 47	sstrdi 4021	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ∘f · 𝐺) ⊆ ℂ)
49	48	ssdifd 4168	. . . . 5 ⊢ (𝜑 → (ran (𝐹 ∘f · 𝐺) ∖ {0}) ⊆ (ℂ ∖ {0}))
50	49	sselda 4008	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
51	3, 6	i1fmullem 25748	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
52	50, 51	syldan 590	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
53		difss 4159	. . . . . 6 ⊢ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺
54		ssfi 9240	. . . . . 6 ⊢ ((ran 𝐺 ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺) → (ran 𝐺 ∖ {0}) ∈ Fin)
55	16, 53, 54	sylancl 585	. . . . 5 ⊢ (𝜑 → (ran 𝐺 ∖ {0}) ∈ Fin)
56		i1fima 25732	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
57	3, 56	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
58		i1fima 25732	. . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
59	6, 58	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
60		inmbl 25596	. . . . . . 7 ⊢ (((◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
61	57, 59, 60	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
62	61	ralrimivw 3156	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
63		finiunmbl 25598	. . . . 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 583	. . . 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 2844	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝑦}) ∈ dom vol)
67		mblvol 25584	. . . 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 25585	. . . . 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 4259	. . . . . . 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 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ∈ dom vol)
75		mblss 25585	. . . . . . 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 25584	. . . . . . . 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 25734	. . . . . . . 8 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
81	79, 80	sylan 579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
82	78, 81	eqeltrrd 2845	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
83		ovolsscl 25540	. . . . . 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 1371	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
85	71, 84	fsumrecl 15782	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
86	52	fveq2d 6924	. . . . 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 25585	. . . . . . . . . 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 725	. . . . . . . 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 3152	. . . . . 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 25555	. . . . . 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 583	. . . . 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 5188	. . . 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 25537	. . . 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 1371	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ∈ ℝ)
97	68, 96	eqeltrd 2844	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f · 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f · 𝐺) “ {𝑦})) ∈ ℝ)
98	12, 45, 66, 97	i1fd 25735	1 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ dom ∫1)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  {csn 4648  ∪ ciun 5015   class class class wbr 5166   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   “ cima 5703   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   ∘_f cof 7712  Fincfn 9003  ℂcc 11182  ℝcr 11183  0cc0 11184   · cmul 11189   ≤ cle 11325   / cdiv 11947  Σcsu 15734  vol*covol 25516  volcvol 25517  ∫₁citg1 25669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xadd 13176  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-xmet 21380  df-met 21381  df-ovol 25518  df-vol 25519  df-mbf 25673  df-itg1 25674

This theorem is referenced by:  mbfmullem2  25779  ftc1anclem3  37655