Theorem i1fadd 25996

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 11264	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
2	1	adantl 487	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
3		i1fadd.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
4		i1ff 25977	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
5	3, 4	syl 18	. . 3 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
6		i1fadd.2	. . . 4 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
7		i1ff 25977	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 18	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 11272	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4172	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 7700	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶ℝ)
13		i1frn 25978	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 25978	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 9295	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
18	14, 16, 17	syl2anc 596	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19		eqid 2761	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20		ovex 7445	. . . . . 6 ⊢ (𝑢 + 𝑣) ∈ V
21	19, 20	fnmpoi 8070	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6794	. . . . 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 233	. . . 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 9289	. . . 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 598	. . 3 ⊢ (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) ∈ Fin)
26		eqid 2761	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = (𝑥 + 𝑦)
27		rspceov 7461	. . . . . . . . 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 1479	. . . . . . . 8 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29		ovex 7445	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) ∈ V
30		eqeq1 2765	. . . . . . . . . 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 3633	. . . . . . . 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 237	. . . . . . 7 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → (𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
34	33	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
35	5	ffnd 6702	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
36		dffn3 6714	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
37	35, 36	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
38	8	ffnd 6702	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
39		dffn3 6714	. . . . . . 7 ⊢ (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
40	38, 39	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐺:ℝ⟶ran 𝐺)
41	34, 37, 40, 10, 10, 11	off 7700	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
42	41	frnd 6710	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
43	19	rnmpo 7545	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
44	42, 43	sseqtrrdi 3972	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
45	25, 44	ssfid 9244	. 2 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ∈ Fin)
46	12	frnd 6710	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ℝ)
47	46	ssdifssd 4094	. . . . . 6 ⊢ (𝜑 → (ran (𝐹 ∘f + 𝐺) ∖ {0}) ⊆ ℝ)
48	47	sselda 3931	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
49	48	recnd 11318	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
50	3, 6	i1faddlem 25994	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
51	49, 50	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
52	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ran 𝐺 ∈ Fin)
53	3	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
54		i1fmbf 25976	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn)
55	53, 54	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ MblFn)
56	5	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
57	12	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 ∘f + 𝐺):ℝ⟶ℝ)
58	57	frnd 6710	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹 ∘f + 𝐺) ⊆ ℝ)
59		eldifi 4078	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
60	59	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
61	58, 60	sseldd 3932	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
62	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
63	62	frnd 6710	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
64	63	sselda 3931	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
65	61, 64	resubcld 11725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 − 𝑧) ∈ ℝ)
66		mbfimasn 25933	. . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦 − 𝑧) ∈ ℝ) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
67	55, 56, 65, 66	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
68	6	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
69		i1fmbf 25976	. . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → 𝐺 ∈ MblFn)
70	68, 69	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ MblFn)
71	8	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
72		mbfimasn 25933	. . . . . . 7 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (◡𝐺 “ {𝑧}) ∈ dom vol)
73	70, 71, 64, 72	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐺 “ {𝑧}) ∈ dom vol)
74		inmbl 25843	. . . . . 6 ⊢ (((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
75	67, 73, 74	syl2anc 596	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
76	75	ralrimiva 3155	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
77		finiunmbl 25845	. . . 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 596	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
79	51, 78	eqeltrd 2861	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) ∈ dom vol)
80		mblvol 25831	. . . 4 ⊢ ((◡(𝐹 ∘f + 𝐺) “ {𝑦}) ∈ dom vol → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})))
81	79, 80	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})))
82		mblss 25832	. . . . 5 ⊢ ((◡(𝐹 ∘f + 𝐺) “ {𝑦}) ∈ dom vol → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) ⊆ ℝ)
83	79, 82	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) ⊆ ℝ)
84		inss1 4182	. . . . . . . 8 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {(𝑦 − 𝑧)})
85	67	adantrr 730	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
86		mblss 25832	. . . . . . . . 9 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol → (◡𝐹 “ {(𝑦 − 𝑧)}) ⊆ ℝ)
87	85, 86	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) ⊆ ℝ)
88		mblvol 25831	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})))
89	85, 88	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})))
90		simprr 785	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → 𝑧 = 0)
91	90	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 𝑧) = (𝑦 − 0))
92	49	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → 𝑦 ∈ ℂ)
93	92	subid1d 11639	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 0) = 𝑦)
94	91, 93	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 𝑧) = 𝑦)
95	94	sneqd 4596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → {(𝑦 − 𝑧)} = {𝑦})
96	95	imaeq2d 6054	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) = (◡𝐹 “ {𝑦}))
97	96	fveq2d 6881	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol‘(◡𝐹 “ {𝑦})))
98		i1fima2sn 25981	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
99	3, 98	sylan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
100	99	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
101	97, 100	eqeltrd 2861	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
102	89, 101	eqeltrrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
103		ovolsscl 25787	. . . . . . . 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 1495	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
105	104	expr 462	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 = 0 → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
106		eldifsn 4748	. . . . . . . 8 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺 ∧ 𝑧 ≠ 0))
107		inss2 4183	. . . . . . . . 9 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧})
108		eldifi 4078	. . . . . . . . . 10 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
109		mblss 25832	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (◡𝐺 “ {𝑧}) ⊆ ℝ)
110	73, 109	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
111	108, 110	sylan2 605	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
112		i1fima 25979	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
113	6, 112	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
114	113	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ∈ dom vol)
115		mblvol 25831	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
116	114, 115	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
117	6	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
118		i1fima2sn 25981	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
119	117, 118	sylan 592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
120	116, 119	eqeltrrd 2862	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
121		ovolsscl 25787	. . . . . . . . 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 1495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
123	106, 122	sylan2br 607	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 ≠ 0)) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
124	123	expr 462	. . . . . 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 3042	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
126	52, 125	fsumrecl 15880	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
127	51	fveq2d 6881	. . . . 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 3942	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
129	128, 125	jca 521	. . . . . . 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 3155	. . . . . 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 25802	. . . . . 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 596	. . . . 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 5127	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
134		ovollecl 25784	. . . 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 1398	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ∈ ℝ)
136	81, 135	eqeltrd 2861	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ∈ ℝ)
137	12, 45, 79, 136	i1fd 25982	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom ∫1)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  {csn 4584  ∪ ciun 4951   class class class wbr 5103   × cxp 5649  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654   Fn wfn 6526  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414   ∘_f cof 7680  Fincfn 8957  ℂcc 11179  ℝcr 11180  0cc0 11181   + caddc 11184   ≤ cle 11325   − cmin 11522  Σcsu 15833  vol*covol 25763  volcvol 25764  MblFncmbf 25915  ∫₁citg1 25916

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-oi 9488  df-dju 9963  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-q 13057  df-rp 13102  df-xadd 13223  df-ioo 13461  df-ico 13463  df-icc 13464  df-fz 13621  df-fzo 13769  df-fl 13912  df-seq 14125  df-exp 14185  df-hash 14455  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-clim 15635  df-sum 15834  df-xmet 21651  df-met 21652  df-ovol 25765  df-vol 25766  df-mbf 25920  df-itg1 25921

This theorem is used by:  itg1addlem4  26000  i1fsub  26009  itg2splitlem  26049  itg2split  26050  itg2addlem  26059  itg2addnc  38560  ftc1anclem3  38581  ftc1anclem5  38583  ftc1anclem8  38586