Theorem i1fadd 24764

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 10885	. . . 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 24745	. . . 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 24745	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 10893	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4149	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 7529	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶ℝ)
13		i1frn 24746	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 24746	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 9015	. . . . 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 2738	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20		ovex 7288	. . . . . 6 ⊢ (𝑢 + 𝑣) ∈ V
21	19, 20	fnmpoi 7883	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6678	. . . . 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 229	. . . 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 9035	. . . 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 2738	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = (𝑥 + 𝑦)
27		rspceov 7302	. . . . . . . . 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 1448	. . . . . . . 8 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29		ovex 7288	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) ∈ V
30		eqeq1 2742	. . . . . . . . . 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 3602	. . . . . . . 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 233	. . . . . . 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 6585	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
36		dffn3 6597	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
37	35, 36	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
38	8	ffnd 6585	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
39		dffn3 6597	. . . . . . 7 ⊢ (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
40	38, 39	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐺:ℝ⟶ran 𝐺)
41	34, 37, 40, 10, 10, 11	off 7529	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
42	41	frnd 6592	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
43	19	rnmpo 7385	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
44	42, 43	sseqtrrdi 3968	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
45	25, 44	ssfid 8971	. 2 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ∈ Fin)
46	12	frnd 6592	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ℝ)
47	46	ssdifssd 4073	. . . . . 6 ⊢ (𝜑 → (ran (𝐹 ∘f + 𝐺) ∖ {0}) ⊆ ℝ)
48	47	sselda 3917	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
49	48	recnd 10934	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
50	3, 6	i1faddlem 24762	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
51	49, 50	syldan 590	. . 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 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
54		i1fmbf 24744	. . . . . . . 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 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
57	12	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 ∘f + 𝐺):ℝ⟶ℝ)
58	57	frnd 6592	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹 ∘f + 𝐺) ⊆ ℝ)
59		eldifi 4057	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
60	59	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
61	58, 60	sseldd 3918	. . . . . . . 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 6592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
64	63	sselda 3917	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
65	61, 64	resubcld 11333	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 − 𝑧) ∈ ℝ)
66		mbfimasn 24701	. . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦 − 𝑧) ∈ ℝ) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
67	55, 56, 65, 66	syl3anc 1369	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
68	6	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
69		i1fmbf 24744	. . . . . . . 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 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
72		mbfimasn 24701	. . . . . . 7 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (◡𝐺 “ {𝑧}) ∈ dom vol)
73	70, 71, 64, 72	syl3anc 1369	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐺 “ {𝑧}) ∈ dom vol)
74		inmbl 24611	. . . . . 6 ⊢ (((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
75	67, 73, 74	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
76	75	ralrimiva 3107	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
77		finiunmbl 24613	. . . 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 583	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
79	51, 78	eqeltrd 2839	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) ∈ dom vol)
80		mblvol 24599	. . . 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 24600	. . . . 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 4159	. . . . . . . 8 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {(𝑦 − 𝑧)})
85	67	adantrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
86		mblss 24600	. . . . . . . . 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 24599	. . . . . . . . . 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 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → 𝑧 = 0)
91	90	oveq2d 7271	. . . . . . . . . . . . . 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 11251	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 0) = 𝑦)
94	91, 93	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 𝑧) = 𝑦)
95	94	sneqd 4570	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → {(𝑦 − 𝑧)} = {𝑦})
96	95	imaeq2d 5958	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) = (◡𝐹 “ {𝑦}))
97	96	fveq2d 6760	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol‘(◡𝐹 “ {𝑦})))
98		i1fima2sn 24749	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
99	3, 98	sylan 579	. . . . . . . . . . 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 2839	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
102	89, 101	eqeltrrd 2840	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
103		ovolsscl 24555	. . . . . . . 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 1464	. . . . . . 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 4717	. . . . . . . 8 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺 ∧ 𝑧 ≠ 0))
107		inss2 4160	. . . . . . . . 9 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧})
108		eldifi 4057	. . . . . . . . . 10 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
109		mblss 24600	. . . . . . . . . . 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 592	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
112		i1fima 24747	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
113	6, 112	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
114	113	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ∈ dom vol)
115		mblvol 24599	. . . . . . . . . . 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 24749	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
119	117, 118	sylan 579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
120	116, 119	eqeltrrd 2840	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
121		ovolsscl 24555	. . . . . . . . 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 1464	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
123	106, 122	sylan2br 594	. . . . . . 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 3030	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
126	52, 125	fsumrecl 15374	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
127	51	fveq2d 6760	. . . . 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 3928	. . . . . . . 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 3107	. . . . . 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 24570	. . . . . 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 583	. . . . 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 5092	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
134		ovollecl 24552	. . . 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 1369	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ∈ ℝ)
136	81, 135	eqeltrd 2839	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ∈ ℝ)
137	12, 45, 79, 136	i1fd 24750	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom ∫1)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  {csn 4558  ∪ ciun 4921   class class class wbr 5070   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ∘_f cof 7509  Fincfn 8691  ℂcc 10800  ℝcr 10801  0cc0 10802   + caddc 10805   ≤ cle 10941   − cmin 11135  Σcsu 15325  vol*covol 24531  volcvol 24532  MblFncmbf 24683  ∫₁citg1 24684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xadd 12778  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-xmet 20503  df-met 20504  df-ovol 24533  df-vol 24534  df-mbf 24688  df-itg1 24689

This theorem is referenced by:  itg1addlem4  24768  itg1addlem4OLD  24769  i1fsub  24778  itg2splitlem  24818  itg2split  24819  itg2addlem  24828  itg2addnc  35758  ftc1anclem3  35779  ftc1anclem5  35781  ftc1anclem8  35784