Theorem i1fadd 23899

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 10355	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
2	1	adantl 475	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
3		i1fadd.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
4		i1ff 23880	. . . 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 23880	. . . 4 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
9		reex 10363	. . . 4 ⊢ ℝ ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
11		inidm 4042	. . 3 ⊢ (ℝ ∩ ℝ) = ℝ
12	2, 5, 8, 10, 10, 11	off 7189	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺):ℝ⟶ℝ)
13		i1frn 23881	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
14	3, 13	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
15		i1frn 23881	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
16	6, 15	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
17		xpfi 8519	. . . . 5 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
18	14, 16, 17	syl2anc 579	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19		eqid 2777	. . . . . 6 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20		ovex 6954	. . . . . 6 ⊢ (𝑢 + 𝑣) ∈ V
21	19, 20	fnmpt2i 7519	. . . . 5 ⊢ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22		dffn4 6372	. . . . 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 222	. . . 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 8540	. . . 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 580	. . 3 ⊢ (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) ∈ Fin)
26		eqid 2777	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = (𝑥 + 𝑦)
27		rspceov 6968	. . . . . . . . 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 1523	. . . . . . . 8 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29		ovex 6954	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) ∈ V
30		eqeq1 2781	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31	30	2rexbidv 3241	. . . . . . . . 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 3557	. . . . . . . 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 226	. . . . . . 7 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → (𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
34	33	adantl 475	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
35	5	ffnd 6292	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℝ)
36		dffn3 6302	. . . . . . 7 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
37	35, 36	sylib 210	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
38	8	ffnd 6292	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ℝ)
39		dffn3 6302	. . . . . . 7 ⊢ (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
40	38, 39	sylib 210	. . . . . 6 ⊢ (𝜑 → 𝐺:ℝ⟶ran 𝐺)
41	34, 37, 40, 10, 10, 11	off 7189	. . . . 5 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
42	41	frnd 6298	. . . 4 ⊢ (𝜑 → ran (𝐹 ∘𝑓 + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
43	19	rnmpt2 7047	. . . 4 ⊢ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹∃𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
44	42, 43	syl6sseqr 3870	. . 3 ⊢ (𝜑 → ran (𝐹 ∘𝑓 + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
45		ssfi 8468	. . 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)
46	25, 44, 45	syl2anc 579	. 2 ⊢ (𝜑 → ran (𝐹 ∘𝑓 + 𝐺) ∈ Fin)
47	12	frnd 6298	. . . . . . 7 ⊢ (𝜑 → ran (𝐹 ∘𝑓 + 𝐺) ⊆ ℝ)
48	47	ssdifssd 3970	. . . . . 6 ⊢ (𝜑 → (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0}) ⊆ ℝ)
49	48	sselda 3820	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
50	49	recnd 10405	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
51	3, 6	i1faddlem 23897	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
52	50, 51	syldan 585	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
53	16	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → ran 𝐺 ∈ Fin)
54	3	ad2antrr 716	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
55		i1fmbf 23879	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn)
56	54, 55	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ MblFn)
57	5	ad2antrr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
58	12	ad2antrr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 ∘𝑓 + 𝐺):ℝ⟶ℝ)
59	58	frnd 6298	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹 ∘𝑓 + 𝐺) ⊆ ℝ)
60		eldifi 3954	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹 ∘𝑓 + 𝐺))
61	60	ad2antlr 717	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹 ∘𝑓 + 𝐺))
62	59, 61	sseldd 3821	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
63	8	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
64	63	frnd 6298	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
65	64	sselda 3820	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
66	62, 65	resubcld 10803	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 − 𝑧) ∈ ℝ)
67		mbfimasn 23836	. . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦 − 𝑧) ∈ ℝ) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
68	56, 57, 66, 67	syl3anc 1439	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
69	6	ad2antrr 716	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
70		i1fmbf 23879	. . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → 𝐺 ∈ MblFn)
71	69, 70	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ MblFn)
72	8	ad2antrr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
73		mbfimasn 23836	. . . . . . 7 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (◡𝐺 “ {𝑧}) ∈ dom vol)
74	71, 72, 65, 73	syl3anc 1439	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐺 “ {𝑧}) ∈ dom vol)
75		inmbl 23746	. . . . . 6 ⊢ (((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
76	68, 74, 75	syl2anc 579	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
77	76	ralrimiva 3147	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
78		finiunmbl 23748	. . . 4 ⊢ ((ran 𝐺 ∈ Fin ∧ ∀𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol) → ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
79	53, 77, 78	syl2anc 579	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
80	52, 79	eqeltrd 2858	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦}) ∈ dom vol)
81		mblvol 23734	. . . 4 ⊢ ((◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦}) ∈ dom vol → (vol‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦})))
82	80, 81	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦})) = (vol*‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦})))
83		mblss 23735	. . . . 5 ⊢ ((◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦}) ∈ dom vol → (◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦}) ⊆ ℝ)
84	80, 83	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦}) ⊆ ℝ)
85		inss1 4052	. . . . . . . . 9 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {(𝑦 − 𝑧)})
86	85	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {(𝑦 − 𝑧)}))
87	68	adantrr 707	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol)
88		mblss 23735	. . . . . . . . 9 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol → (◡𝐹 “ {(𝑦 − 𝑧)}) ⊆ ℝ)
89	87, 88	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) ⊆ ℝ)
90		mblvol 23734	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∈ dom vol → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})))
91	87, 90	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})))
92		simprr 763	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → 𝑧 = 0)
93	92	oveq2d 6938	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 𝑧) = (𝑦 − 0))
94	50	adantr 474	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → 𝑦 ∈ ℂ)
95	94	subid1d 10723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 0) = 𝑦)
96	93, 95	eqtrd 2813	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (𝑦 − 𝑧) = 𝑦)
97	96	sneqd 4409	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → {(𝑦 − 𝑧)} = {𝑦})
98	97	imaeq2d 5720	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (◡𝐹 “ {(𝑦 − 𝑧)}) = (◡𝐹 “ {𝑦}))
99	98	fveq2d 6450	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) = (vol‘(◡𝐹 “ {𝑦})))
100		i1fima2sn 23884	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
101	3, 100	sylan 575	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
102	101	adantr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {𝑦})) ∈ ℝ)
103	99, 102	eqeltrd 2858	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
104	91, 103	eqeltrrd 2859	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol*‘(◡𝐹 “ {(𝑦 − 𝑧)})) ∈ ℝ)
105		ovolsscl 23690	. . . . . . . 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)
106	86, 89, 104, 105	syl3anc 1439	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
107	106	expr 450	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 = 0 → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
108		eldifsn 4549	. . . . . . . 8 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺 ∧ 𝑧 ≠ 0))
109		inss2 4053	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧})
110	109	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧}))
111		eldifi 3954	. . . . . . . . . 10 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
112		mblss 23735	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (◡𝐺 “ {𝑧}) ⊆ ℝ)
113	74, 112	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
114	111, 113	sylan2 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
115		i1fima 23882	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
116	6, 115	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
117	116	ad2antrr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ∈ dom vol)
118		mblvol 23734	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑧}) ∈ dom vol → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
119	117, 118	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) = (vol*‘(◡𝐺 “ {𝑧})))
120	6	adantr 474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
121		i1fima2sn 23884	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
122	120, 121	sylan 575	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) ∈ ℝ)
123	119, 122	eqeltrrd 2859	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ)
124		ovolsscl 23690	. . . . . . . . 9 ⊢ ((((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧}) ∧ (◡𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(◡𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
125	110, 114, 123, 124	syl3anc 1439	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
126	108, 125	sylan2br 588	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 ≠ 0)) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
127	126	expr 450	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 ≠ 0 → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
128	107, 127	pm2.61dne 3055	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
129	53, 128	fsumrecl 14872	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
130	52	fveq2d 6450	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦})) = (vol*‘∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
131	109, 113	syl5ss 3831	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ)
132	131, 128	jca 507	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
133	132	ralrimiva 3147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺(((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ))
134		ovolfiniun 23705	. . . . . 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}))))
135	53, 133, 134	syl2anc 579	. . . . 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}))))
136	130, 135	eqbrtrd 4908	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((◡𝐹 “ {(𝑦 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
137		ovollecl 23687	. . . 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)
138	84, 129, 136, 137	syl3anc 1439	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (vol*‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦})) ∈ ℝ)
139	82, 138	eqeltrd 2858	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {0})) → (vol‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑦})) ∈ ℝ)
140	12, 46, 80, 139	i1fd 23885	1 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ dom ∫1)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  {cab 2762   ≠ wne 2968  ∀wral 3089  ∃wrex 3090  Vcvv 3397   ∖ cdif 3788   ∩ cin 3790   ⊆ wss 3791  {csn 4397  ∪ ciun 4753   class class class wbr 4886   × cxp 5353  ^◡ccnv 5354  dom cdm 5355  ran crn 5356   “ cima 5358   Fn wfn 6130  ⟶wf 6131  –onto→wfo 6133  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924   ∘_𝑓 cof 7172  Fincfn 8241  ℂcc 10270  ℝcr 10271  0cc0 10272   + caddc 10275   ≤ cle 10412   − cmin 10606  Σcsu 14824  vol*covol 23666  volcvol 23667  MblFncmbf 23818  ∫₁citg1 23819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-q 12096  df-rp 12138  df-xadd 12258  df-ioo 12491  df-ico 12493  df-icc 12494  df-fz 12644  df-fzo 12785  df-fl 12912  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-sum 14825  df-xmet 20135  df-met 20136  df-ovol 23668  df-vol 23669  df-mbf 23823  df-itg1 23824

This theorem is referenced by:  itg1addlem4  23903  i1fsub  23912  itg2splitlem  23952  itg2split  23953  itg2addlem  23962  itg2addnc  34073  ftc1anclem3  34096  ftc1anclem5  34098  ftc1anclem8  34101