Theorem i1fmul 23463

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 10021	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
2	1	adantl 482	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
3		i1fadd.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
4		i1ff 23443	. . . 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 23443	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 10027	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 3822	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 6912	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺):ℝ⟶ℝ)
13		i1frn 23444	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 23444	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 8231	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
18	14, 16, 17	syl2anc 693	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19		eqid 2622	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
20		ovex 6678	. . . . . 6 ⊢ (𝑢 · 𝑣) ∈ V
21	19, 20	fnmpt2i 7239	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6121	. . . . 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 220	. . . 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 8252	. . . 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 694	. . 3 ⊢ (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
26		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) = (𝑥 · 𝑦)
27		rspceov 6692	. . . . . . . . 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 1413	. . . . . . . 8 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
29		ovex 6678	. . . . . . . . 9 ⊢ (𝑥 · 𝑦) ∈ V
30		eqeq1 2626	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 · 𝑦) → (𝑤 = (𝑢 · 𝑣) ↔ (𝑥 · 𝑦) = (𝑢 · 𝑣)))
31	30	2rexbidv 3057	. . . . . . . . 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 3350	. . . . . . . 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 224	. . . . . . 7 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
34	33	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
35		ffn 6045	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
36	5, 35	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
37		dffn3 6054	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
38	36, 37	sylib 208	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
39		ffn 6045	. . . . . . . 8 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
40	8, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
41		dffn3 6054	. . . . . . 7 ⊢ (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
42	40, 41	sylib 208	. . . . . 6 ⊢ (𝜑 → 𝐺:ℝ⟶ran 𝐺)
43	34, 38, 42, 10, 10, 11	off 6912	. . . . 5 ⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
44		frn 6053	. . . . 5 ⊢ ((𝐹 ∘𝑓 · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)} → ran (𝐹 ∘𝑓 · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
45	43, 44	syl 17	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘𝑓 · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
46	19	rnmpt2 6770	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)}
47	45, 46	syl6sseqr 3652	. . 3 ⊢ (𝜑 → ran (𝐹 ∘𝑓 · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
48		ssfi 8180	. . 3 ⊢ ((ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin ∧ ran (𝐹 ∘𝑓 · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))) → ran (𝐹 ∘𝑓 · 𝐺) ∈ Fin)
49	25, 47, 48	syl2anc 693	. 2 ⊢ (𝜑 → ran (𝐹 ∘𝑓 · 𝐺) ∈ Fin)
50		frn 6053	. . . . . . . 8 ⊢ ((𝐹 ∘𝑓 · 𝐺):ℝ⟶ℝ → ran (𝐹 ∘𝑓 · 𝐺) ⊆ ℝ)
51	12, 50	syl 17	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘𝑓 · 𝐺) ⊆ ℝ)
52		ax-resscn 9993	. . . . . . 7 ⊢ ℝ ⊆ ℂ
53	51, 52	syl6ss 3615	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ∘𝑓 · 𝐺) ⊆ ℂ)
54	53	ssdifd 3746	. . . . 5 ⊢ (𝜑 → (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0}) ⊆ (ℂ ∖ {0}))
55	54	sselda 3603	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
56	3, 6	i1fmullem 23461	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
57	55, 56	syldan 487	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → (◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
58		difss 3737	. . . . . 6 ⊢ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺
59		ssfi 8180	. . . . . 6 ⊢ ((ran 𝐺 ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺) → (ran 𝐺 ∖ {0}) ∈ Fin)
60	16, 58, 59	sylancl 694	. . . . 5 ⊢ (𝜑 → (ran 𝐺 ∖ {0}) ∈ Fin)
61		i1fima 23445	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
62	3, 61	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
63		i1fima 23445	. . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
64	6, 63	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
65		inmbl 23310	. . . . . . 7 ⊢ (((◡𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
66	62, 64, 65	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
67	66	ralrimivw 2967	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
68		finiunmbl 23312	. . . . 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)
69	60, 67, 68	syl2anc 693	. . . 4 ⊢ (𝜑 → ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
70	69	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → ∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
71	57, 70	eqeltrd 2701	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → (◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦}) ∈ dom vol)
72		mblvol 23298	. . . 4 ⊢ ((◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦}) ∈ dom vol → (vol‘(◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦})))
73	71, 72	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦})))
74		mblss 23299	. . . . 5 ⊢ ((◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦}) ∈ dom vol → (◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦}) ⊆ ℝ)
75	71, 74	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → (◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦}) ⊆ ℝ)
76	60	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → (ran 𝐺 ∖ {0}) ∈ Fin)
77		inss2 3834	. . . . . . 7 ⊢ ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧})
78	77	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧}))
79	64	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ∈ dom vol)
80		mblss 23299	. . . . . . 7 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (◡𝐺 “ {𝑧}) ⊆ ℝ)
81	79, 80	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
82		mblvol 23298	. . . . . . . 8 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
83	79, 82	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
84	6	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
85		i1fima2sn 23447	. . . . . . . 8 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
86	84, 85	sylan 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
87	83, 86	eqeltrrd 2702	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
88		ovolsscl 23254	. . . . . 6 ⊢ ((((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧}) ∧ (◡𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
89	78, 81, 87, 88	syl3anc 1326	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
90	76, 89	fsumrecl 14465	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
91	57	fveq2d 6195	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦})) = (vol*‘∪ 𝑧 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
92		mblss 23299	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
93	66, 92	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
94	93	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
95	94, 89	jca 554	. . . . . . 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))
96	95	ralrimiva 2966	. . . . . 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))
97		ovolfiniun 23269	. . . . . 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}))))
98	76, 96, 97	syl2anc 693	. . . . 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}))))
99	91, 98	eqbrtrd 4675	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((◡𝐹 “ {(𝑦 / 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
100		ovollecl 23251	. . . 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)
101	75, 90, 99, 100	syl3anc 1326	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦})) ∈ ℝ)
102	73, 101	eqeltrd 2701	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 · 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘𝑓 · 𝐺) “ {𝑦})) ∈ ℝ)
103	12, 49, 71, 102	i1fd 23448	1 ⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺) ∈ dom ∫1)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ∪ ciun 4520   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ∘_𝑓 cof 6895  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936   · cmul 9941   ≤ cle 10075   / cdiv 10684  Σcsu 14416  vol*covol 23231  volcvol 23232  ∫₁citg1 23384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389

This theorem is referenced by:  mbfmullem2  23491  ftc1anclem3  33487