Theorem i1fadd 25731

Description: The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)

Hypotheses

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

Assertion

Ref	Expression
i1fadd	⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom ∫1)

Proof of Theorem i1fadd

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

Step	Hyp	Ref	Expression
1		readdcl 11239	. . . 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 25712	. . . 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 25712	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 11247	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4226	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 7716	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶ℝ)
13		i1frn 25713	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 25713	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 9359	. . . . 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 2736	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20		ovex 7465	. . . . . 6 ⊢ (𝑢 + 𝑣) ∈ V
21	19, 20	fnmpoi 8096	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6825	. . . . 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 9352	. . . 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 2736	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = (𝑥 + 𝑦)
27		rspceov 7481	. . . . . . . . 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 1451	. . . . . . . 8 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29		ovex 7465	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) ∈ V
30		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31	30	2rexbidv 3221	. . . . . . . . 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 3678	. . . . . . . 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 6736	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
36		dffn3 6747	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
37	35, 36	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
38	8	ffnd 6736	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
39		dffn3 6747	. . . . . . 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 7716	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
42	41	frnd 6743	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
43	19	rnmpo 7567	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
44	42, 43	sseqtrrdi 4024	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
45	25, 44	ssfid 9302	. 2 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ∈ Fin)
46	12	frnd 6743	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ℝ)
47	46	ssdifssd 4146	. . . . . 6 ⊢ (𝜑 → (ran (𝐹 ∘f + 𝐺) ∖ {0}) ⊆ ℝ)
48	47	sselda 3982	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
49	48	recnd 11290	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
50	3, 6	i1faddlem 25729	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
51	49, 50	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
52	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ran 𝐺 ∈ Fin)
53	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
54		i1fmbf 25711	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn)
55	53, 54	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ MblFn)
56	5	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
57	12	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 ∘f + 𝐺):ℝ⟶ℝ)
58	57	frnd 6743	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹 ∘f + 𝐺) ⊆ ℝ)
59		eldifi 4130	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
60	59	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
61	58, 60	sseldd 3983	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
62	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
63	62	frnd 6743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
64	63	sselda 3982	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
65	61, 64	resubcld 11692	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 − 𝑧) ∈ ℝ)
66		mbfimasn 25668	. . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦 − 𝑧) ∈ ℝ) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
67	55, 56, 65, 66	syl3anc 1372	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
68	6	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
69		i1fmbf 25711	. . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → 𝐺 ∈ MblFn)
70	68, 69	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ MblFn)
71	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
72		mbfimasn 25668	. . . . . . 7 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (◡𝐺 “ {𝑧}) ∈ dom vol)
73	70, 71, 64, 72	syl3anc 1372	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐺 “ {𝑧}) ∈ dom vol)
74		inmbl 25578	. . . . . 6 ⊢ (((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
75	67, 73, 74	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
76	75	ralrimiva 3145	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
77		finiunmbl 25580	. . . 4 ⊢ ((ran 𝐺 ∈ Fin ∧ ∀𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol) → ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
78	52, 76, 77	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
79	51, 78	eqeltrd 2840	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) ∈ dom vol)
80		mblvol 25566	. . . 4 ⊢ ((◡(𝐹 ∘f + 𝐺) “ {𝑦}) ∈ dom vol → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})))
81	79, 80	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})))
82		mblss 25567	. . . . 5 ⊢ ((◡(𝐹 ∘f + 𝐺) “ {𝑦}) ∈ dom vol → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) ⊆ ℝ)
83	79, 82	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) ⊆ ℝ)
84		inss1 4236	. . . . . . . 8 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {(𝑦 − 𝑧)})
85	67	adantrr 717	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
86		mblss 25567	. . . . . . . . 9 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol → (◡𝐹 “ {(𝑦 − 𝑧)}) ⊆ ℝ)
87	85, 86	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) ⊆ ℝ)
88		mblvol 25566	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})))
89	85, 88	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})))
90		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → 𝑧 = 0)
91	90	oveq2d 7448	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 𝑧) = (𝑦 − 0))
92	49	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → 𝑦 ∈ ℂ)
93	92	subid1d 11610	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 0) = 𝑦)
94	91, 93	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 𝑧) = 𝑦)
95	94	sneqd 4637	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → {(𝑦 − 𝑧)} = {𝑦})
96	95	imaeq2d 6077	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) = (◡𝐹 “ {𝑦}))
97	96	fveq2d 6909	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol‘(◡𝐹 “ {𝑦})))
98		i1fima2sn 25716	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
99	3, 98	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
100	99	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
101	97, 100	eqeltrd 2840	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
102	89, 101	eqeltrrd 2841	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
103		ovolsscl 25522	. . . . . . . 8 ⊢ ((((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {(𝑦 − 𝑧)}) ∧ (◡𝐹 “ {(𝑦 − 𝑧)}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
104	84, 87, 102, 103	mp3an2i 1467	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
105	104	expr 456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 = 0 → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
106		eldifsn 4785	. . . . . . . 8 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺 ∧ 𝑧 ≠ 0))
107		inss2 4237	. . . . . . . . 9 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧})
108		eldifi 4130	. . . . . . . . . 10 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
109		mblss 25567	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (◡𝐺 “ {𝑧}) ⊆ ℝ)
110	73, 109	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
111	108, 110	sylan2 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
112		i1fima 25714	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
113	6, 112	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
114	113	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ∈ dom vol)
115		mblvol 25566	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
116	114, 115	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
117	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
118		i1fima2sn 25716	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
119	117, 118	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
120	116, 119	eqeltrrd 2841	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
121		ovolsscl 25522	. . . . . . . . 9 ⊢ ((((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧}) ∧ (◡𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
122	107, 111, 120, 121	mp3an2i 1467	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
123	106, 122	sylan2br 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 ≠ 0)) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
124	123	expr 456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 ≠ 0 → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
125	105, 124	pm2.61dne 3027	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
126	52, 125	fsumrecl 15771	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
127	51	fveq2d 6909	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) = (vol*‘∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
128	107, 110	sstrid 3994	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
129	128, 125	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
130	129	ralrimiva 3145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺(((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
131		ovolfiniun 25537	. . . . . 6 ⊢ ((ran 𝐺 ∈ Fin ∧ ∀𝑧 ∈ ran 𝐺(((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)) → (vol*‘∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
132	52, 130, 131	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
133	127, 132	eqbrtrd 5164	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
134		ovollecl 25519	. . . 4 ⊢ (((◡(𝐹 ∘f + 𝐺) “ {𝑦}) ⊆ ℝ ∧ Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ ∧ (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ∈ ℝ)
135	83, 126, 133, 134	syl3anc 1372	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ∈ ℝ)
136	81, 135	eqeltrd 2840	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ∈ ℝ)
137	12, 45, 79, 136	i1fd 25717	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom ∫1)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2713   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∖ cdif 3947   ∩ cin 3949   ⊆ wss 3950  {csn 4625  ∪ ciun 4990   class class class wbr 5142   × cxp 5682  ^◡ccnv 5683  dom cdm 5684  ran crn 5685   “ cima 5687   Fn wfn 6555  ⟶wf 6556  –onto→wfo 6558  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434   ∘_f cof 7696  Fincfn 8986  ℂcc 11154  ℝcr 11155  0cc0 11156   + caddc 11159   ≤ cle 11297   − cmin 11493  Σcsu 15723  vol*covol 25498  volcvol 25499  MblFncmbf 25650  ∫₁citg1 25651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-xadd 13156  df-ioo 13392  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-fl 13833  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-sum 15724  df-xmet 21358  df-met 21359  df-ovol 25500  df-vol 25501  df-mbf 25655  df-itg1 25656

This theorem is referenced by:  itg1addlem4  25735  i1fsub  25744  itg2splitlem  25784  itg2split  25785  itg2addlem  25794  itg2addnc  37682  ftc1anclem3  37703  ftc1anclem5  37705  ftc1anclem8  37708