hgt750lemg - Mathbox for Thierry Arnoux

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Theorem hgt750lemg 34155

Description: Lemma for the statement 7.50 of [Helfgott] p. 69. Applying a permutation 𝑇 to the three factors of a product does not change the result. (Contributed by Thierry Arnoux, 1-Jan-2022.)

**Hypotheses**
Ref	Expression
hgt750lemg.f	⊢ 𝐹 = (𝑐 ∈ 𝑅 ↦ (𝑐 ∘ 𝑇))
hgt750lemg.t	⊢ (𝜑 → 𝑇:(0..^3)–1-1-onto→(0..^3))
hgt750lemg.n	⊢ (𝜑 → 𝑁:(0..^3)⟶ℕ)
hgt750lemg.l	⊢ (𝜑 → 𝐿:ℕ⟶ℝ)
hgt750lemg.1	⊢ (𝜑 → 𝑁 ∈ 𝑅)

**Assertion**
Ref	Expression
hgt750lemg	⊢ (𝜑 → ((𝐿‘((𝐹‘𝑁)‘0)) · ((𝐿‘((𝐹‘𝑁)‘1)) · (𝐿‘((𝐹‘𝑁)‘2)))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2)))))

Distinct variable groups: 𝑁,𝑐 𝑅,𝑐 𝑇,𝑐 𝜑,𝑐 Allowed substitution hints: 𝐹(𝑐) 𝐿(𝑐)

Proof of Theorem hgt750lemg Dummy variables 𝑏 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6886	. . . . 5 ⊢ (𝑎 = (𝑇‘𝑏) → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘(𝑇‘𝑏))))
2		tpfi 9319	. . . . . 6 ⊢ {0, 1, 2} ∈ Fin
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → {0, 1, 2} ∈ Fin)
4		hgt750lemg.t	. . . . . 6 ⊢ (𝜑 → 𝑇:(0..^3)–1-1-onto→(0..^3))
5		fzo0to3tp 13715	. . . . . . 7 ⊢ (0..^3) = {0, 1, 2}
6		f1oeq23 6814	. . . . . . 7 ⊢ (((0..^3) = {0, 1, 2} ∧ (0..^3) = {0, 1, 2}) → (𝑇:(0..^3)–1-1-onto→(0..^3) ↔ 𝑇:{0, 1, 2}–1-1-onto→{0, 1, 2}))
7	5, 5, 6	mp2an 689	. . . . . 6 ⊢ (𝑇:(0..^3)–1-1-onto→(0..^3) ↔ 𝑇:{0, 1, 2}–1-1-onto→{0, 1, 2})
8	4, 7	sylib 217	. . . . 5 ⊢ (𝜑 → 𝑇:{0, 1, 2}–1-1-onto→{0, 1, 2})
9		eqidd 2725	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝑇‘𝑏) = (𝑇‘𝑏))
10		hgt750lemg.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿:ℕ⟶ℝ)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝐿:ℕ⟶ℝ)
12		hgt750lemg.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁:(0..^3)⟶ℕ)
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝑁:(0..^3)⟶ℕ)
14		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝑎 ∈ {0, 1, 2})
15	14, 5	eleqtrrdi 2836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → 𝑎 ∈ (0..^3))
16	13, 15	ffvelcdmd 7077	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → (𝑁‘𝑎) ∈ ℕ)
17	11, 16	ffvelcdmd 7077	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → (𝐿‘(𝑁‘𝑎)) ∈ ℝ)
18	17	recnd 11239	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ {0, 1, 2}) → (𝐿‘(𝑁‘𝑎)) ∈ ℂ)
19	1, 3, 8, 9, 18	fprodf1o 15887	. . . 4 ⊢ (𝜑 → ∏𝑎 ∈ {0, 1, 2} (𝐿‘(𝑁‘𝑎)) = ∏𝑏 ∈ {0, 1, 2} (𝐿‘(𝑁‘(𝑇‘𝑏))))
20		hgt750lemg.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑐 ∈ 𝑅 ↦ (𝑐 ∘ 𝑇))
21	20	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑐 ∈ 𝑅 ↦ (𝑐 ∘ 𝑇)))
22		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 = 𝑁) → 𝑐 = 𝑁)
23	22	coeq1d 5851	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 = 𝑁) → (𝑐 ∘ 𝑇) = (𝑁 ∘ 𝑇))
24		hgt750lemg.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ 𝑅)
25		f1of 6823	. . . . . . . . . . . . 13 ⊢ (𝑇:(0..^3)–1-1-onto→(0..^3) → 𝑇:(0..^3)⟶(0..^3))
26	4, 25	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇:(0..^3)⟶(0..^3))
27		ovexd 7436	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^3) ∈ V)
28	26, 27	fexd 7220	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ V)
29		coexg 7913	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑅 ∧ 𝑇 ∈ V) → (𝑁 ∘ 𝑇) ∈ V)
30	24, 28, 29	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑇) ∈ V)
31	21, 23, 24, 30	fvmptd 6995	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑁) = (𝑁 ∘ 𝑇))
32	31	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝐹‘𝑁) = (𝑁 ∘ 𝑇))
33	32	fveq1d 6883	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → ((𝐹‘𝑁)‘𝑏) = ((𝑁 ∘ 𝑇)‘𝑏))
34		f1ofun 6825	. . . . . . . . . 10 ⊢ (𝑇:(0..^3)–1-1-onto→(0..^3) → Fun 𝑇)
35	4, 34	syl 17	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝑇)
36	35	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → Fun 𝑇)
37		f1odm 6827	. . . . . . . . . . 11 ⊢ (𝑇:{0, 1, 2}–1-1-onto→{0, 1, 2} → dom 𝑇 = {0, 1, 2})
38	8, 37	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑇 = {0, 1, 2})
39	38	eleq2d 2811	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∈ dom 𝑇 ↔ 𝑏 ∈ {0, 1, 2}))
40	39	biimpar 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → 𝑏 ∈ dom 𝑇)
41		fvco 6979	. . . . . . . 8 ⊢ ((Fun 𝑇 ∧ 𝑏 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘𝑏) = (𝑁‘(𝑇‘𝑏)))
42	36, 40, 41	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → ((𝑁 ∘ 𝑇)‘𝑏) = (𝑁‘(𝑇‘𝑏)))
43	33, 42	eqtr2d 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝑁‘(𝑇‘𝑏)) = ((𝐹‘𝑁)‘𝑏))
44	43	fveq2d 6885	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {0, 1, 2}) → (𝐿‘(𝑁‘(𝑇‘𝑏))) = (𝐿‘((𝐹‘𝑁)‘𝑏)))
45	44	prodeq2dv 15864	. . . 4 ⊢ (𝜑 → ∏𝑏 ∈ {0, 1, 2} (𝐿‘(𝑁‘(𝑇‘𝑏))) = ∏𝑏 ∈ {0, 1, 2} (𝐿‘((𝐹‘𝑁)‘𝑏)))
46	19, 45	eqtr2d 2765	. . 3 ⊢ (𝜑 → ∏𝑏 ∈ {0, 1, 2} (𝐿‘((𝐹‘𝑁)‘𝑏)) = ∏𝑎 ∈ {0, 1, 2} (𝐿‘(𝑁‘𝑎)))
47		2fveq3 6886	. . . 4 ⊢ (𝑏 = 0 → (𝐿‘((𝐹‘𝑁)‘𝑏)) = (𝐿‘((𝐹‘𝑁)‘0)))
48		2fveq3 6886	. . . 4 ⊢ (𝑏 = 1 → (𝐿‘((𝐹‘𝑁)‘𝑏)) = (𝐿‘((𝐹‘𝑁)‘1)))
49		c0ex 11205	. . . . 5 ⊢ 0 ∈ V
50	49	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
51		1ex 11207	. . . . 5 ⊢ 1 ∈ V
52	51	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ V)
53	31	fveq1d 6883	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑁)‘0) = ((𝑁 ∘ 𝑇)‘0))
54	49	tpid1 4764	. . . . . . . . . 10 ⊢ 0 ∈ {0, 1, 2}
55	54, 38	eleqtrrid 2832	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ dom 𝑇)
56		fvco 6979	. . . . . . . . 9 ⊢ ((Fun 𝑇 ∧ 0 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘0) = (𝑁‘(𝑇‘0)))
57	35, 55, 56	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 ∘ 𝑇)‘0) = (𝑁‘(𝑇‘0)))
58	53, 57	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁)‘0) = (𝑁‘(𝑇‘0)))
59	54, 5	eleqtrri 2824	. . . . . . . . . 10 ⊢ 0 ∈ (0..^3)
60	59	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0..^3))
61	26, 60	ffvelcdmd 7077	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘0) ∈ (0..^3))
62	12, 61	ffvelcdmd 7077	. . . . . . 7 ⊢ (𝜑 → (𝑁‘(𝑇‘0)) ∈ ℕ)
63	58, 62	eqeltrd 2825	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑁)‘0) ∈ ℕ)
64	10, 63	ffvelcdmd 7077	. . . . 5 ⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘0)) ∈ ℝ)
65	64	recnd 11239	. . . 4 ⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘0)) ∈ ℂ)
66	31	fveq1d 6883	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑁)‘1) = ((𝑁 ∘ 𝑇)‘1))
67	51	tpid2 4766	. . . . . . . . . 10 ⊢ 1 ∈ {0, 1, 2}
68	67, 38	eleqtrrid 2832	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ dom 𝑇)
69		fvco 6979	. . . . . . . . 9 ⊢ ((Fun 𝑇 ∧ 1 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘1) = (𝑁‘(𝑇‘1)))
70	35, 68, 69	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 ∘ 𝑇)‘1) = (𝑁‘(𝑇‘1)))
71	66, 70	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁)‘1) = (𝑁‘(𝑇‘1)))
72	67, 5	eleqtrri 2824	. . . . . . . . . 10 ⊢ 1 ∈ (0..^3)
73	72	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ (0..^3))
74	26, 73	ffvelcdmd 7077	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘1) ∈ (0..^3))
75	12, 74	ffvelcdmd 7077	. . . . . . 7 ⊢ (𝜑 → (𝑁‘(𝑇‘1)) ∈ ℕ)
76	71, 75	eqeltrd 2825	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑁)‘1) ∈ ℕ)
77	10, 76	ffvelcdmd 7077	. . . . 5 ⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘1)) ∈ ℝ)
78	77	recnd 11239	. . . 4 ⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘1)) ∈ ℂ)
79		0ne1 12280	. . . . 5 ⊢ 0 ≠ 1
80	79	a1i 11	. . . 4 ⊢ (𝜑 → 0 ≠ 1)
81		2fveq3 6886	. . . 4 ⊢ (𝑏 = 2 → (𝐿‘((𝐹‘𝑁)‘𝑏)) = (𝐿‘((𝐹‘𝑁)‘2)))
82		2ex 12286	. . . . 5 ⊢ 2 ∈ V
83	82	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ V)
84	31	fveq1d 6883	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑁)‘2) = ((𝑁 ∘ 𝑇)‘2))
85	82	tpid3 4769	. . . . . . . . . 10 ⊢ 2 ∈ {0, 1, 2}
86	85, 38	eleqtrrid 2832	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ dom 𝑇)
87		fvco 6979	. . . . . . . . 9 ⊢ ((Fun 𝑇 ∧ 2 ∈ dom 𝑇) → ((𝑁 ∘ 𝑇)‘2) = (𝑁‘(𝑇‘2)))
88	35, 86, 87	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 ∘ 𝑇)‘2) = (𝑁‘(𝑇‘2)))
89	84, 88	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁)‘2) = (𝑁‘(𝑇‘2)))
90	85, 5	eleqtrri 2824	. . . . . . . . . 10 ⊢ 2 ∈ (0..^3)
91	90	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ (0..^3))
92	26, 91	ffvelcdmd 7077	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘2) ∈ (0..^3))
93	12, 92	ffvelcdmd 7077	. . . . . . 7 ⊢ (𝜑 → (𝑁‘(𝑇‘2)) ∈ ℕ)
94	89, 93	eqeltrd 2825	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑁)‘2) ∈ ℕ)
95	10, 94	ffvelcdmd 7077	. . . . 5 ⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘2)) ∈ ℝ)
96	95	recnd 11239	. . . 4 ⊢ (𝜑 → (𝐿‘((𝐹‘𝑁)‘2)) ∈ ℂ)
97		0ne2 12416	. . . . 5 ⊢ 0 ≠ 2
98	97	a1i 11	. . . 4 ⊢ (𝜑 → 0 ≠ 2)
99		1ne2 12417	. . . . 5 ⊢ 1 ≠ 2
100	99	a1i 11	. . . 4 ⊢ (𝜑 → 1 ≠ 2)
101	47, 48, 50, 52, 65, 78, 80, 81, 83, 96, 98, 100	prodtp 32500	. . 3 ⊢ (𝜑 → ∏𝑏 ∈ {0, 1, 2} (𝐿‘((𝐹‘𝑁)‘𝑏)) = (((𝐿‘((𝐹‘𝑁)‘0)) · (𝐿‘((𝐹‘𝑁)‘1))) · (𝐿‘((𝐹‘𝑁)‘2))))
102		2fveq3 6886	. . . 4 ⊢ (𝑎 = 0 → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘0)))
103		2fveq3 6886	. . . 4 ⊢ (𝑎 = 1 → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘1)))
104	12, 60	ffvelcdmd 7077	. . . . . 6 ⊢ (𝜑 → (𝑁‘0) ∈ ℕ)
105	10, 104	ffvelcdmd 7077	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝑁‘0)) ∈ ℝ)
106	105	recnd 11239	. . . 4 ⊢ (𝜑 → (𝐿‘(𝑁‘0)) ∈ ℂ)
107	12, 73	ffvelcdmd 7077	. . . . . 6 ⊢ (𝜑 → (𝑁‘1) ∈ ℕ)
108	10, 107	ffvelcdmd 7077	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝑁‘1)) ∈ ℝ)
109	108	recnd 11239	. . . 4 ⊢ (𝜑 → (𝐿‘(𝑁‘1)) ∈ ℂ)
110		2fveq3 6886	. . . 4 ⊢ (𝑎 = 2 → (𝐿‘(𝑁‘𝑎)) = (𝐿‘(𝑁‘2)))
111	12, 91	ffvelcdmd 7077	. . . . . 6 ⊢ (𝜑 → (𝑁‘2) ∈ ℕ)
112	10, 111	ffvelcdmd 7077	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝑁‘2)) ∈ ℝ)
113	112	recnd 11239	. . . 4 ⊢ (𝜑 → (𝐿‘(𝑁‘2)) ∈ ℂ)
114	102, 103, 50, 52, 106, 109, 80, 110, 83, 113, 98, 100	prodtp 32500	. . 3 ⊢ (𝜑 → ∏𝑎 ∈ {0, 1, 2} (𝐿‘(𝑁‘𝑎)) = (((𝐿‘(𝑁‘0)) · (𝐿‘(𝑁‘1))) · (𝐿‘(𝑁‘2))))
115	46, 101, 114	3eqtr3d 2772	. 2 ⊢ (𝜑 → (((𝐿‘((𝐹‘𝑁)‘0)) · (𝐿‘((𝐹‘𝑁)‘1))) · (𝐿‘((𝐹‘𝑁)‘2))) = (((𝐿‘(𝑁‘0)) · (𝐿‘(𝑁‘1))) · (𝐿‘(𝑁‘2))))
116	65, 78, 96	mulassd 11234	. 2 ⊢ (𝜑 → (((𝐿‘((𝐹‘𝑁)‘0)) · (𝐿‘((𝐹‘𝑁)‘1))) · (𝐿‘((𝐹‘𝑁)‘2))) = ((𝐿‘((𝐹‘𝑁)‘0)) · ((𝐿‘((𝐹‘𝑁)‘1)) · (𝐿‘((𝐹‘𝑁)‘2)))))
117	106, 109, 113	mulassd 11234	. 2 ⊢ (𝜑 → (((𝐿‘(𝑁‘0)) · (𝐿‘(𝑁‘1))) · (𝐿‘(𝑁‘2))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2)))))
118	115, 116, 117	3eqtr3d 2772	1 ⊢ (𝜑 → ((𝐿‘((𝐹‘𝑁)‘0)) · ((𝐿‘((𝐹‘𝑁)‘1)) · (𝐿‘((𝐹‘𝑁)‘2)))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 {ctp 4624 ↦ cmpt 5221 dom cdm 5666 ∘ ccom 5670 Fun wfun 6527 ⟶wf 6529 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7401 Fincfn 8935 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 ℕcn 12209 2c2 12264 3c3 12265 ..^cfzo 13624 ∏cprod 15846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-prod 15847

This theorem is referenced by: hgt750lema 34158

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