Theorem i1fadd 25672

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 11112	. . . 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 25653	. . . 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 25653	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 11120	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4168	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 7642	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶ℝ)
13		i1frn 25654	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 25654	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 9223	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
18	14, 16, 17	syl2anc 585	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19		eqid 2737	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20		ovex 7393	. . . . . 6 ⊢ (𝑢 + 𝑣) ∈ V
21	19, 20	fnmpoi 8016	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6752	. . . . 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 9216	. . . 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 587	. . 3 ⊢ (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) ∈ Fin)
26		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = (𝑥 + 𝑦)
27		rspceov 7409	. . . . . . . . 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 1453	. . . . . . . 8 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29		ovex 7393	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) ∈ V
30		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31	30	2rexbidv 3203	. . . . . . . . 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 3623	. . . . . . . 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 6663	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
36		dffn3 6674	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
37	35, 36	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
38	8	ffnd 6663	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
39		dffn3 6674	. . . . . . 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 7642	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
42	41	frnd 6670	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
43	19	rnmpo 7493	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
44	42, 43	sseqtrrdi 3964	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
45	25, 44	ssfid 9172	. 2 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ∈ Fin)
46	12	frnd 6670	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ℝ)
47	46	ssdifssd 4088	. . . . . 6 ⊢ (𝜑 → (ran (𝐹 ∘f + 𝐺) ∖ {0}) ⊆ ℝ)
48	47	sselda 3922	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
49	48	recnd 11164	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
50	3, 6	i1faddlem 25670	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
51	49, 50	syldan 592	. . 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 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
54		i1fmbf 25652	. . . . . . . 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 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
57	12	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 ∘f + 𝐺):ℝ⟶ℝ)
58	57	frnd 6670	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹 ∘f + 𝐺) ⊆ ℝ)
59		eldifi 4072	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
60	59	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
61	58, 60	sseldd 3923	. . . . . . . 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 6670	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
64	63	sselda 3922	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
65	61, 64	resubcld 11569	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 − 𝑧) ∈ ℝ)
66		mbfimasn 25609	. . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦 − 𝑧) ∈ ℝ) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
67	55, 56, 65, 66	syl3anc 1374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
68	6	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
69		i1fmbf 25652	. . . . . . . 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 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
72		mbfimasn 25609	. . . . . . 7 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (◡𝐺 “ {𝑧}) ∈ dom vol)
73	70, 71, 64, 72	syl3anc 1374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐺 “ {𝑧}) ∈ dom vol)
74		inmbl 25519	. . . . . 6 ⊢ (((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
75	67, 73, 74	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
76	75	ralrimiva 3130	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
77		finiunmbl 25521	. . . 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 585	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
79	51, 78	eqeltrd 2837	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (◡(𝐹 ∘f + 𝐺) “ {𝑦}) ∈ dom vol)
80		mblvol 25507	. . . 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 25508	. . . . 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 4178	. . . . . . . 8 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {(𝑦 − 𝑧)})
85	67	adantrr 718	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
86		mblss 25508	. . . . . . . . 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 25507	. . . . . . . . . 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 773	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → 𝑧 = 0)
91	90	oveq2d 7376	. . . . . . . . . . . . . 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 11485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 0) = 𝑦)
94	91, 93	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 𝑧) = 𝑦)
95	94	sneqd 4580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → {(𝑦 − 𝑧)} = {𝑦})
96	95	imaeq2d 6019	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) = (◡𝐹 “ {𝑦}))
97	96	fveq2d 6838	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol‘(◡𝐹 “ {𝑦})))
98		i1fima2sn 25657	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
99	3, 98	sylan 581	. . . . . . . . . . 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 2837	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
102	89, 101	eqeltrrd 2838	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
103		ovolsscl 25463	. . . . . . . 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 1469	. . . . . . 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 4730	. . . . . . . 8 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺 ∧ 𝑧 ≠ 0))
107		inss2 4179	. . . . . . . . 9 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧})
108		eldifi 4072	. . . . . . . . . 10 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
109		mblss 25508	. . . . . . . . . . 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 594	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
112		i1fima 25655	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
113	6, 112	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
114	113	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ∈ dom vol)
115		mblvol 25507	. . . . . . . . . . 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 25657	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
119	117, 118	sylan 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
120	116, 119	eqeltrrd 2838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
121		ovolsscl 25463	. . . . . . . . 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 1469	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
123	106, 122	sylan2br 596	. . . . . . 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 3019	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
126	52, 125	fsumrecl 15687	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
127	51	fveq2d 6838	. . . . 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 3934	. . . . . . . 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 3130	. . . . . 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 25478	. . . . . 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 585	. . . . 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 5108	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
134		ovollecl 25460	. . . 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 1374	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ∈ ℝ)
136	81, 135	eqeltrd 2837	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑦})) ∈ ℝ)
137	12, 45, 79, 136	i1fd 25658	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom ∫1)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  {csn 4568  ∪ ciun 4934   class class class wbr 5086   × cxp 5622  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362   ∘_f cof 7622  Fincfn 8886  ℂcc 11027  ℝcr 11028  0cc0 11029   + caddc 11032   ≤ cle 11171   − cmin 11368  Σcsu 15639  vol*covol 25439  volcvol 25440  MblFncmbf 25591  ∫₁citg1 25592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-oi 9418  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890  df-rp 12934  df-xadd 13055  df-ioo 13293  df-ico 13295  df-icc 13296  df-fz 13453  df-fzo 13600  df-fl 13742  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-sum 15640  df-xmet 21337  df-met 21338  df-ovol 25441  df-vol 25442  df-mbf 25596  df-itg1 25597

This theorem is referenced by:  itg1addlem4  25676  i1fsub  25685  itg2splitlem  25725  itg2split  25726  itg2addlem  25735  itg2addnc  38009  ftc1anclem3  38030  ftc1anclem5  38032  ftc1anclem8  38035