Theorem itg1addlem4OLD 24769

Description: Obsolete version of itg1addlem4 24768. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
i1fadd.1	⊢ (𝜑 → 𝐹 ∈ dom ∫1)
i1fadd.2	⊢ (𝜑 → 𝐺 ∈ dom ∫1)
itg1add.3	⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))
itg1add.4	⊢ 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))

Assertion

Ref	Expression
itg1addlem4OLD	⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))

Distinct variable groups:   𝑖,𝑗,𝑦,𝑧   𝑦,𝐼   𝑦,𝑃,𝑧   𝑖,𝐹,𝑗,𝑦,𝑧   𝑖,𝐺,𝑗,𝑦,𝑧   𝜑,𝑖,𝑗,𝑦,𝑧

Allowed substitution hints:   𝑃(𝑖,𝑗)   𝐼(𝑧,𝑖,𝑗)

Proof of Theorem itg1addlem4OLD

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

Step	Hyp	Ref	Expression
1		i1fadd.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
2		i1fadd.2	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
3	1, 2	i1fadd 24764	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom ∫1)
4		i1frn 24746	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
6		i1frn 24746	. . . . . . . 8 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
7	2, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
8		xpfi 9015	. . . . . . 7 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
9	5, 7, 8	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
10		ax-addf 10881	. . . . . . . . . 10 ⊢ + :(ℂ × ℂ)⟶ℂ
11		ffn 6584	. . . . . . . . . 10 ⊢ ( + :(ℂ × ℂ)⟶ℂ → + Fn (ℂ × ℂ))
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ + Fn (ℂ × ℂ)
13		i1ff 24745	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
14	1, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
15	14	frnd 6592	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
16		ax-resscn 10859	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
17	15, 16	sstrdi 3929	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
18		i1ff 24745	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
19	2, 18	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
20	19	frnd 6592	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ)
21	20, 16	sstrdi 3929	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐺 ⊆ ℂ)
22		xpss12 5595	. . . . . . . . . 10 ⊢ ((ran 𝐹 ⊆ ℂ ∧ ran 𝐺 ⊆ ℂ) → (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ))
23	17, 21, 22	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ))
24		fnssres 6539	. . . . . . . . 9 ⊢ (( + Fn (ℂ × ℂ) ∧ (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ)) → ( + ↾ (ran 𝐹 × ran 𝐺)) Fn (ran 𝐹 × ran 𝐺))
25	12, 23, 24	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → ( + ↾ (ran 𝐹 × ran 𝐺)) Fn (ran 𝐹 × ran 𝐺))
26		itg1add.4	. . . . . . . . 9 ⊢ 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
27	26	fneq1i 6514	. . . . . . . 8 ⊢ (𝑃 Fn (ran 𝐹 × ran 𝐺) ↔ ( + ↾ (ran 𝐹 × ran 𝐺)) Fn (ran 𝐹 × ran 𝐺))
28	25, 27	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 𝑃 Fn (ran 𝐹 × ran 𝐺))
29		dffn4 6678	. . . . . . 7 ⊢ (𝑃 Fn (ran 𝐹 × ran 𝐺) ↔ 𝑃:(ran 𝐹 × ran 𝐺)–onto→ran 𝑃)
30	28, 29	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝑃:(ran 𝐹 × ran 𝐺)–onto→ran 𝑃)
31		fofi 9035	. . . . . 6 ⊢ (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ 𝑃:(ran 𝐹 × ran 𝐺)–onto→ran 𝑃) → ran 𝑃 ∈ Fin)
32	9, 30, 31	syl2anc 583	. . . . 5 ⊢ (𝜑 → ran 𝑃 ∈ Fin)
33		difss 4062	. . . . 5 ⊢ (ran 𝑃 ∖ {0}) ⊆ ran 𝑃
34		ssfi 8918	. . . . 5 ⊢ ((ran 𝑃 ∈ Fin ∧ (ran 𝑃 ∖ {0}) ⊆ ran 𝑃) → (ran 𝑃 ∖ {0}) ∈ Fin)
35	32, 33, 34	sylancl 585	. . . 4 ⊢ (𝜑 → (ran 𝑃 ∖ {0}) ∈ Fin)
36		ffun 6587	. . . . . . . . . . 11 ⊢ ( + :(ℂ × ℂ)⟶ℂ → Fun + )
37	10, 36	ax-mp 5	. . . . . . . . . 10 ⊢ Fun +
38	10	fdmi 6596	. . . . . . . . . . 11 ⊢ dom + = (ℂ × ℂ)
39	23, 38	sseqtrrdi 3968	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ dom + )
40		funfvima2 7089	. . . . . . . . . 10 ⊢ ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
41	37, 39, 40	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
42		opelxpi 5617	. . . . . . . . 9 ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) → ⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺))
43	41, 42	impel 505	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺)))
44		df-ov 7258	. . . . . . . 8 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
45	26	rneqi 5835	. . . . . . . . 9 ⊢ ran 𝑃 = ran ( + ↾ (ran 𝐹 × ran 𝐺))
46		df-ima 5593	. . . . . . . . 9 ⊢ ( + “ (ran 𝐹 × ran 𝐺)) = ran ( + ↾ (ran 𝐹 × ran 𝐺))
47	45, 46	eqtr4i 2769	. . . . . . . 8 ⊢ ran 𝑃 = ( + “ (ran 𝐹 × ran 𝐺))
48	43, 44, 47	3eltr4g 2856	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ ran 𝑃)
49	14	ffnd 6585	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn ℝ)
50		dffn3 6597	. . . . . . . 8 ⊢ (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
51	49, 50	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ran 𝐹)
52	19	ffnd 6585	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn ℝ)
53		dffn3 6597	. . . . . . . 8 ⊢ (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
54	52, 53	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℝ⟶ran 𝐺)
55		reex 10893	. . . . . . . 8 ⊢ ℝ ∈ V
56	55	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
57		inidm 4149	. . . . . . 7 ⊢ (ℝ ∩ ℝ) = ℝ
58	48, 51, 54, 56, 56, 57	off 7529	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):ℝ⟶ran 𝑃)
59	58	frnd 6592	. . . . 5 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ran 𝑃)
60	59	ssdifd 4071	. . . 4 ⊢ (𝜑 → (ran (𝐹 ∘f + 𝐺) ∖ {0}) ⊆ (ran 𝑃 ∖ {0}))
61	15	sselda 3917	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
62	20	sselda 3917	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
63	61, 62	anim12dan 618	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐺)) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
64		readdcl 10885	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 + 𝑧) ∈ ℝ)
65	63, 64	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐺)) → (𝑦 + 𝑧) ∈ ℝ)
66	65	ralrimivva 3114	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ)
67		funimassov 7427	. . . . . . . 8 ⊢ ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ ↔ ∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
68	37, 39, 67	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ ↔ ∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
69	66, 68	mpbird 256	. . . . . 6 ⊢ (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ)
70	47, 69	eqsstrid 3965	. . . . 5 ⊢ (𝜑 → ran 𝑃 ⊆ ℝ)
71	70	ssdifd 4071	. . . 4 ⊢ (𝜑 → (ran 𝑃 ∖ {0}) ⊆ (ℝ ∖ {0}))
72		itg1val2 24753	. . . 4 ⊢ (((𝐹 ∘f + 𝐺) ∈ dom ∫1 ∧ ((ran 𝑃 ∖ {0}) ∈ Fin ∧ (ran (𝐹 ∘f + 𝐺) ∖ {0}) ⊆ (ran 𝑃 ∖ {0}) ∧ (ran 𝑃 ∖ {0}) ⊆ (ℝ ∖ {0}))) → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑤}))))
73	3, 35, 60, 71, 72	syl13anc 1370	. . 3 ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑤}))))
74	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝐺:ℝ⟶ℝ)
75	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → ran 𝐺 ∈ Fin)
76		inss2 4160	. . . . . . . . 9 ⊢ ((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧})
77	76	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧}))
78		i1fima 24747	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {(𝑤 − 𝑧)}) ∈ dom vol)
79	1, 78	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐹 “ {(𝑤 − 𝑧)}) ∈ dom vol)
80		i1fima 24747	. . . . . . . . . . 11 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
81	2, 80	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
82		inmbl 24611	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {(𝑤 − 𝑧)}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
83	79, 81, 82	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
84	83	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
85	33, 70	sstrid 3928	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝑃 ∖ {0}) ⊆ ℝ)
86	85	sselda 3917	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝑤 ∈ ℝ)
87	86	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ∈ ℝ)
88	62	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
89	87, 88	resubcld 11333	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑤 − 𝑧) ∈ ℝ)
90	87	recnd 10934	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ∈ ℂ)
91	88	recnd 10934	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
92	90, 91	npcand 11266	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤 − 𝑧) + 𝑧) = 𝑤)
93		eldifsni 4720	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (ran 𝑃 ∖ {0}) → 𝑤 ≠ 0)
94	93	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ≠ 0)
95	92, 94	eqnetrd 3010	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤 − 𝑧) + 𝑧) ≠ 0)
96		oveq12 7264	. . . . . . . . . . . . 13 ⊢ (((𝑤 − 𝑧) = 0 ∧ 𝑧 = 0) → ((𝑤 − 𝑧) + 𝑧) = (0 + 0))
97		00id 11080	. . . . . . . . . . . . 13 ⊢ (0 + 0) = 0
98	96, 97	eqtrdi 2795	. . . . . . . . . . . 12 ⊢ (((𝑤 − 𝑧) = 0 ∧ 𝑧 = 0) → ((𝑤 − 𝑧) + 𝑧) = 0)
99	98	necon3ai 2967	. . . . . . . . . . 11 ⊢ (((𝑤 − 𝑧) + 𝑧) ≠ 0 → ¬ ((𝑤 − 𝑧) = 0 ∧ 𝑧 = 0))
100	95, 99	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ¬ ((𝑤 − 𝑧) = 0 ∧ 𝑧 = 0))
101		itg1add.3	. . . . . . . . . . 11 ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))
102	1, 2, 101	itg1addlem3 24767	. . . . . . . . . 10 ⊢ ((((𝑤 − 𝑧) ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ ((𝑤 − 𝑧) = 0 ∧ 𝑧 = 0)) → ((𝑤 − 𝑧)𝐼𝑧) = (vol‘((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
103	89, 88, 100, 102	syl21anc 834	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤 − 𝑧)𝐼𝑧) = (vol‘((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
104	1, 2, 101	itg1addlem2 24766	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ)
105	104	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
106	105, 89, 88	fovrnd 7422	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤 − 𝑧)𝐼𝑧) ∈ ℝ)
107	103, 106	eqeltrrd 2840	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
108	74, 75, 77, 84, 107	itg1addlem1 24761	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
109	86	recnd 10934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝑤 ∈ ℂ)
110	1, 2	i1faddlem 24762	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝑤}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
111	109, 110	syldan 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (◡(𝐹 ∘f + 𝐺) “ {𝑤}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧})))
112	111	fveq2d 6760	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑤})) = (vol‘∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
113	103	sumeq2dv 15343	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → Σ𝑧 ∈ ran 𝐺((𝑤 − 𝑧)𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((◡𝐹 “ {(𝑤 − 𝑧)}) ∩ (◡𝐺 “ {𝑧}))))
114	108, 112, 113	3eqtr4d 2788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑤})) = Σ𝑧 ∈ ran 𝐺((𝑤 − 𝑧)𝐼𝑧))
115	114	oveq2d 7271	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑤}))) = (𝑤 · Σ𝑧 ∈ ran 𝐺((𝑤 − 𝑧)𝐼𝑧)))
116	106	recnd 10934	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤 − 𝑧)𝐼𝑧) ∈ ℂ)
117	75, 109, 116	fsummulc2 15424	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · Σ𝑧 ∈ ran 𝐺((𝑤 − 𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
118	115, 117	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑤}))) = Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
119	118	sumeq2dv 15343	. . 3 ⊢ (𝜑 → Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘(◡(𝐹 ∘f + 𝐺) “ {𝑤}))) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
120	90, 116	mulcld 10926	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑤 · ((𝑤 − 𝑧)𝐼𝑧)) ∈ ℂ)
121	120	anasss 466	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ (ran 𝑃 ∖ {0}) ∧ 𝑧 ∈ ran 𝐺)) → (𝑤 · ((𝑤 − 𝑧)𝐼𝑧)) ∈ ℂ)
122	35, 7, 121	fsumcom 15415	. . 3 ⊢ (𝜑 → Σ𝑤 ∈ (ran 𝑃 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
123	73, 119, 122	3eqtrd 2782	. 2 ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
124		oveq1 7262	. . . . . . 7 ⊢ (𝑦 = (𝑤 − 𝑧) → (𝑦 + 𝑧) = ((𝑤 − 𝑧) + 𝑧))
125		oveq1 7262	. . . . . . 7 ⊢ (𝑦 = (𝑤 − 𝑧) → (𝑦𝐼𝑧) = ((𝑤 − 𝑧)𝐼𝑧))
126	124, 125	oveq12d 7273	. . . . . 6 ⊢ (𝑦 = (𝑤 − 𝑧) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (((𝑤 − 𝑧) + 𝑧) · ((𝑤 − 𝑧)𝐼𝑧)))
127	32	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ran 𝑃 ∈ Fin)
128	70	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ran 𝑃 ⊆ ℝ)
129	128	sselda 3917	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑣 ∈ ℝ)
130	62	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑧 ∈ ℝ)
131	129, 130	resubcld 11333	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → (𝑣 − 𝑧) ∈ ℝ)
132	131	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 → (𝑣 − 𝑧) ∈ ℝ))
133	129	recnd 10934	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑣 ∈ ℂ)
134	133	adantrr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) → 𝑣 ∈ ℂ)
135	70	sselda 3917	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → 𝑦 ∈ ℝ)
136	135	ad2ant2rl 745	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) → 𝑦 ∈ ℝ)
137	136	recnd 10934	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) → 𝑦 ∈ ℂ)
138	62	recnd 10934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
139	138	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) → 𝑧 ∈ ℂ)
140	134, 137, 139	subcan2ad 11307	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) → ((𝑣 − 𝑧) = (𝑦 − 𝑧) ↔ 𝑣 = 𝑦))
141	140	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ((𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃) → ((𝑣 − 𝑧) = (𝑦 − 𝑧) ↔ 𝑣 = 𝑦)))
142	132, 141	dom2lem 8735	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃–1-1→ℝ)
143		f1f1orn 6711	. . . . . . 7 ⊢ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃–1-1→ℝ → (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃–1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)))
144	142, 143	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃–1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)))
145		oveq1 7262	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 − 𝑧) = (𝑤 − 𝑧))
146		eqid 2738	. . . . . . . 8 ⊢ (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) = (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))
147		ovex 7288	. . . . . . . 8 ⊢ (𝑤 − 𝑧) ∈ V
148	145, 146, 147	fvmpt 6857	. . . . . . 7 ⊢ (𝑤 ∈ ran 𝑃 → ((𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))‘𝑤) = (𝑤 − 𝑧))
149	148	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → ((𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))‘𝑤) = (𝑤 − 𝑧))
150		f1f 6654	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃–1-1→ℝ → (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃⟶ℝ)
151		frn 6591	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃⟶ℝ → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) ⊆ ℝ)
152	142, 150, 151	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) ⊆ ℝ)
153	152	sselda 3917	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))) → 𝑦 ∈ ℝ)
154	62	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))) → 𝑧 ∈ ℝ)
155	153, 154	readdcld 10935	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))) → (𝑦 + 𝑧) ∈ ℝ)
156	104	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))) → 𝐼:(ℝ × ℝ)⟶ℝ)
157	156, 153, 154	fovrnd 7422	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))) → (𝑦𝐼𝑧) ∈ ℝ)
158	155, 157	remulcld 10936	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℝ)
159	158	recnd 10934	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
160	126, 127, 144, 149, 159	fsumf1o 15363	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(((𝑤 − 𝑧) + 𝑧) · ((𝑤 − 𝑧)𝐼𝑧)))
161	128	sselda 3917	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑤 ∈ ℝ)
162	161	recnd 10934	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑤 ∈ ℂ)
163	138	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑧 ∈ ℂ)
164	162, 163	npcand 11266	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → ((𝑤 − 𝑧) + 𝑧) = 𝑤)
165	164	oveq1d 7270	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → (((𝑤 − 𝑧) + 𝑧) · ((𝑤 − 𝑧)𝐼𝑧)) = (𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
166	165	sumeq2dv 15343	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → Σ𝑤 ∈ ran 𝑃(((𝑤 − 𝑧) + 𝑧) · ((𝑤 − 𝑧)𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
167	160, 166	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
168	39	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (ran 𝐹 × ran 𝐺) ⊆ dom + )
169		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
170		simplr 765	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐺)
171	169, 170	opelxpd 5618	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺))
172		funfvima2 7089	. . . . . . . . . . . 12 ⊢ ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
173	37, 172	mpan 686	. . . . . . . . . . 11 ⊢ ((ran 𝐹 × ran 𝐺) ⊆ dom + → (⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
174	168, 171, 173	sylc 65	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺)))
175		df-ov 7258	. . . . . . . . . 10 ⊢ (𝑦 + 𝑧) = ( + ‘⟨𝑦, 𝑧⟩)
176	174, 175, 47	3eltr4g 2856	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 + 𝑧) ∈ ran 𝑃)
177	61	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
178	177	recnd 10934	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
179	138	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
180	178, 179	pncand 11263	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) − 𝑧) = 𝑦)
181	180	eqcomd 2744	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 = ((𝑦 + 𝑧) − 𝑧))
182		oveq1 7262	. . . . . . . . . 10 ⊢ (𝑣 = (𝑦 + 𝑧) → (𝑣 − 𝑧) = ((𝑦 + 𝑧) − 𝑧))
183	182	rspceeqv 3567	. . . . . . . . 9 ⊢ (((𝑦 + 𝑧) ∈ ran 𝑃 ∧ 𝑦 = ((𝑦 + 𝑧) − 𝑧)) → ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣 − 𝑧))
184	176, 181, 183	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣 − 𝑧))
185	184	ralrimiva 3107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ∀𝑦 ∈ ran 𝐹∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣 − 𝑧))
186		ssabral 3992	. . . . . . 7 ⊢ (ran 𝐹 ⊆ {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣 − 𝑧)} ↔ ∀𝑦 ∈ ran 𝐹∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣 − 𝑧))
187	185, 186	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣 − 𝑧)})
188	146	rnmpt 5853	. . . . . 6 ⊢ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) = {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣 − 𝑧)}
189	187, 188	sseqtrrdi 3968	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)))
190	62	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
191	177, 190	readdcld 10935	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 + 𝑧) ∈ ℝ)
192	104	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
193	192, 177, 190	fovrnd 7422	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
194	191, 193	remulcld 10936	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℝ)
195	194	recnd 10934	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
196	152	ssdifd 4071	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) ∖ ran 𝐹) ⊆ (ℝ ∖ ran 𝐹))
197	196	sselda 3917	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) ∖ ran 𝐹)) → 𝑦 ∈ (ℝ ∖ ran 𝐹))
198		eldifi 4057	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (ℝ ∖ ran 𝐹) → 𝑦 ∈ ℝ)
199	198	ad2antrl 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → 𝑦 ∈ ℝ)
200	62	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → 𝑧 ∈ ℝ)
201		simprr 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
202	1, 2, 101	itg1addlem3 24767	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
203	199, 200, 201, 202	syl21anc 834	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦𝐼𝑧) = (vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
204		inss1 4159	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {𝑦})
205		eldifn 4058	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (ℝ ∖ ran 𝐹) → ¬ 𝑦 ∈ ran 𝐹)
206	205	ad2antrl 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ 𝑦 ∈ ran 𝐹)
207		vex 3426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑣 ∈ V
208	207	eliniseg 5991	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ V → (𝑣 ∈ (◡𝐹 “ {𝑦}) ↔ 𝑣𝐹𝑦))
209	208	elv 3428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ (◡𝐹 “ {𝑦}) ↔ 𝑣𝐹𝑦)
210		vex 3426	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ∈ V
211	207, 210	brelrn 5840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣𝐹𝑦 → 𝑦 ∈ ran 𝐹)
212	209, 211	sylbi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ (◡𝐹 “ {𝑦}) → 𝑦 ∈ ran 𝐹)
213	206, 212	nsyl 140	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ 𝑣 ∈ (◡𝐹 “ {𝑦}))
214	213	pm2.21d 121	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑣 ∈ (◡𝐹 “ {𝑦}) → 𝑣 ∈ ∅))
215	214	ssrdv 3923	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (◡𝐹 “ {𝑦}) ⊆ ∅)
216	204, 215	sstrid 3928	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ⊆ ∅)
217		ss0 4329	. . . . . . . . . . . . . 14 ⊢ (((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ⊆ ∅ → ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) = ∅)
218	216, 217	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) = ∅)
219	218	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))) = (vol‘∅))
220		0mbl 24608	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ dom vol
221		mblvol 24599	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
222	220, 221	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol‘∅) = (vol*‘∅)
223		ovol0 24562	. . . . . . . . . . . . 13 ⊢ (vol*‘∅) = 0
224	222, 223	eqtri 2766	. . . . . . . . . . . 12 ⊢ (vol‘∅) = 0
225	219, 224	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))) = 0)
226	203, 225	eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦𝐼𝑧) = 0)
227	226	oveq2d 7271	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 + 𝑧) · 0))
228	199, 200	readdcld 10935	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦 + 𝑧) ∈ ℝ)
229	228	recnd 10934	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦 + 𝑧) ∈ ℂ)
230	229	mul01d 11104	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · 0) = 0)
231	227, 230	eqtrd 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
232	231	expr 456	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ ∖ ran 𝐹)) → (¬ (𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0))
233		oveq12 7264	. . . . . . . . . 10 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦 + 𝑧) = (0 + 0))
234	233, 97	eqtrdi 2795	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦 + 𝑧) = 0)
235		oveq12 7264	. . . . . . . . . 10 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦𝐼𝑧) = (0𝐼0))
236		0re 10908	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
237		iftrue 4462	. . . . . . . . . . . 12 ⊢ ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = 0)
238		c0ex 10900	. . . . . . . . . . . 12 ⊢ 0 ∈ V
239	237, 101, 238	ovmpoa 7406	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → (0𝐼0) = 0)
240	236, 236, 239	mp2an 688	. . . . . . . . . 10 ⊢ (0𝐼0) = 0
241	235, 240	eqtrdi 2795	. . . . . . . . 9 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦𝐼𝑧) = 0)
242	234, 241	oveq12d 7273	. . . . . . . 8 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (0 · 0))
243		0cn 10898	. . . . . . . . 9 ⊢ 0 ∈ ℂ
244	243	mul01i 11095	. . . . . . . 8 ⊢ (0 · 0) = 0
245	242, 244	eqtrdi 2795	. . . . . . 7 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
246	232, 245	pm2.61d2 181	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ ∖ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
247	197, 246	syldan 590	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) ∖ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
248		f1ofo 6707	. . . . . . 7 ⊢ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃–1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) → (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃–onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)))
249	144, 248	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃–onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)))
250		fofi 9035	. . . . . 6 ⊢ ((ran 𝑃 ∈ Fin ∧ (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)):ran 𝑃–onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) ∈ Fin)
251	127, 249, 250	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧)) ∈ Fin)
252	189, 195, 247, 251	fsumss 15365	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 − 𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
253	33	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (ran 𝑃 ∖ {0}) ⊆ ran 𝑃)
254	120	an32s 648	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · ((𝑤 − 𝑧)𝐼𝑧)) ∈ ℂ)
255		dfin4 4198	. . . . . . . 8 ⊢ (ran 𝑃 ∩ {0}) = (ran 𝑃 ∖ (ran 𝑃 ∖ {0}))
256		inss2 4160	. . . . . . . 8 ⊢ (ran 𝑃 ∩ {0}) ⊆ {0}
257	255, 256	eqsstrri 3952	. . . . . . 7 ⊢ (ran 𝑃 ∖ (ran 𝑃 ∖ {0})) ⊆ {0}
258	257	sseli 3913	. . . . . 6 ⊢ (𝑤 ∈ (ran 𝑃 ∖ (ran 𝑃 ∖ {0})) → 𝑤 ∈ {0})
259		elsni 4575	. . . . . . . . 9 ⊢ (𝑤 ∈ {0} → 𝑤 = 0)
260	259	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑤 = 0)
261	260	oveq1d 7270	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤 · ((𝑤 − 𝑧)𝐼𝑧)) = (0 · ((𝑤 − 𝑧)𝐼𝑧)))
262	104	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝐼:(ℝ × ℝ)⟶ℝ)
263	260, 236	eqeltrdi 2847	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑤 ∈ ℝ)
264	62	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑧 ∈ ℝ)
265	263, 264	resubcld 11333	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤 − 𝑧) ∈ ℝ)
266	262, 265, 264	fovrnd 7422	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → ((𝑤 − 𝑧)𝐼𝑧) ∈ ℝ)
267	266	recnd 10934	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → ((𝑤 − 𝑧)𝐼𝑧) ∈ ℂ)
268	267	mul02d 11103	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (0 · ((𝑤 − 𝑧)𝐼𝑧)) = 0)
269	261, 268	eqtrd 2778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤 · ((𝑤 − 𝑧)𝐼𝑧)) = 0)
270	258, 269	sylan2 592	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ (ran 𝑃 ∖ (ran 𝑃 ∖ {0}))) → (𝑤 · ((𝑤 − 𝑧)𝐼𝑧)) = 0)
271	253, 254, 270, 127	fsumss 15365	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
272	167, 252, 271	3eqtr4d 2788	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
273	272	sumeq2dv 15343	. 2 ⊢ (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤 − 𝑧)𝐼𝑧)))
274	195	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
275	7, 5, 274	fsumcom 15415	. 2 ⊢ (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
276	123, 273, 275	3eqtr2d 2784	1 ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ifcif 4456  {csn 4558  ⟨cop 4564  ∪ ciun 4921   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ∘_f cof 7509  Fincfn 8691  ℂcc 10800  ℝcr 10801  0cc0 10802   + caddc 10805   · cmul 10807   − cmin 11135  Σcsu 15325  vol*covol 24531  volcvol 24532  ∫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  ax-addf 10881

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-disj 5036  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: (None)