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Theorem ecgrtg 28231

Description: The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)

**Hypotheses**
Ref	Expression
ecgrtg.1	⊢ (𝜑 → 𝑁 ∈ ℕ)
ecgrtg.2	⊢ 𝑃 = (Base‘(EEG‘𝑁))
ecgrtg.3	⊢ − = (dist‘(EEG‘𝑁))
ecgrtg.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
ecgrtg.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
ecgrtg.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
ecgrtg.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)

**Assertion**
Ref	Expression
ecgrtg	⊢ (𝜑 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷)))

Proof of Theorem ecgrtg Dummy variables 𝑥 𝑖 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecgrtg.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
2		ecgrtg.1	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3		eengbas 28229	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
5		ecgrtg.2	. . . . 5 ⊢ 𝑃 = (Base‘(EEG‘𝑁))
6	4, 5	eqtr4di 2791	. . . 4 ⊢ (𝜑 → (𝔼‘𝑁) = 𝑃)
7	1, 6	eleqtrrd 2837	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁))
8		ecgrtg.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8, 6	eleqtrrd 2837	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁))
10		ecgrtg.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10, 6	eleqtrrd 2837	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁))
12		ecgrtg.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	12, 6	eleqtrrd 2837	. . 3 ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁))
14		brcgr 28148	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
15	7, 9, 11, 13, 14	syl22anc 838	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
16		ecgrtg.3	. . . . . 6 ⊢ − = (dist‘(EEG‘𝑁))
17		dsid 17328	. . . . . . 7 ⊢ dist = Slot (dist‘ndx)
18		fvexd 6904	. . . . . . 7 ⊢ (𝜑 → (EEG‘𝑁) ∈ V)
19		eengstr 28228	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct ⟨1, ;17⟩)
20	2, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (EEG‘𝑁) Struct ⟨1, ;17⟩)
21		structn0fun 17081	. . . . . . . . 9 ⊢ ((EEG‘𝑁) Struct ⟨1, ;17⟩ → Fun ((EEG‘𝑁) ∖ {∅}))
22	20, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun ((EEG‘𝑁) ∖ {∅}))
23		structcnvcnv 17083	. . . . . . . . . 10 ⊢ ((EEG‘𝑁) Struct ⟨1, ;17⟩ → ^◡^◡(EEG‘𝑁) = ((EEG‘𝑁) ∖ {∅}))
24	20, 23	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ^◡^◡(EEG‘𝑁) = ((EEG‘𝑁) ∖ {∅}))
25	24	funeqd 6568	. . . . . . . 8 ⊢ (𝜑 → (Fun ^◡^◡(EEG‘𝑁) ↔ Fun ((EEG‘𝑁) ∖ {∅})))
26	22, 25	mpbird 257	. . . . . . 7 ⊢ (𝜑 → Fun ^◡^◡(EEG‘𝑁))
27		opex 5464	. . . . . . . . . 10 ⊢ ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩ ∈ V
28	27	prid2 4767	. . . . . . . . 9 ⊢ ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩ ∈ {⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩}
29		elun1 4176	. . . . . . . . 9 ⊢ (⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩ ∈ {⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} → ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩ ∈ ({⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))
30	28, 29	ax-mp 5	. . . . . . . 8 ⊢ ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩ ∈ ({⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩})
31		eengv 28227	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) = ({⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))
32	2, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → (EEG‘𝑁) = ({⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))
33	30, 32	eleqtrrid 2841	. . . . . . 7 ⊢ (𝜑 → ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩ ∈ (EEG‘𝑁))
34		fvex 6902	. . . . . . . . 9 ⊢ (𝔼‘𝑁) ∈ V
35	34, 34	mpoex 8063	. . . . . . . 8 ⊢ (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2)) ∈ V
36	35	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2)) ∈ V)
37	17, 18, 26, 33, 36	strfv2d 17132	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2)) = (dist‘(EEG‘𝑁)))
38	16, 37	eqtr4id 2792	. . . . 5 ⊢ (𝜑 → − = (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2)))
39		simplrl 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑥 = 𝐴)
40	39	fveq1d 6891	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) = (𝐴‘𝑖))
41		simplrr 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑦 = 𝐵)
42	41	fveq1d 6891	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) = (𝐵‘𝑖))
43	40, 42	oveq12d 7424	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) − (𝑦‘𝑖)) = ((𝐴‘𝑖) − (𝐵‘𝑖)))
44	43	oveq1d 7421	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) − (𝑦‘𝑖))↑2) = (((𝐴‘𝑖) − (𝐵‘𝑖))↑2))
45	44	sumeq2dv 15646	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2))
46		sumex 15631	. . . . . 6 ⊢ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) ∈ V
47	46	a1i 11	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) ∈ V)
48	38, 45, 7, 9, 47	ovmpod 7557	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) = Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2))
49	48	eqcomd 2739	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = (𝐴 − 𝐵))
50		simplrl 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑥 = 𝐶)
51	50	fveq1d 6891	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) = (𝐶‘𝑖))
52		simplrr 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑦 = 𝐷)
53	52	fveq1d 6891	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) = (𝐷‘𝑖))
54	51, 53	oveq12d 7424	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) − (𝑦‘𝑖)) = ((𝐶‘𝑖) − (𝐷‘𝑖)))
55	54	oveq1d 7421	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) − (𝑦‘𝑖))↑2) = (((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
56	55	sumeq2dv 15646	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) → Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
57		sumex 15631	. . . . . 6 ⊢ Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) ∈ V
58	57	a1i 11	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) ∈ V)
59	38, 56, 11, 13, 58	ovmpod 7557	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐷) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
60	59	eqcomd 2739	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) = (𝐶 − 𝐷))
61	49, 60	eqeq12d 2749	. 2 ⊢ (𝜑 → (Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷)))
62	15, 61	bitrd 279	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 ∖ cdif 3945 ∪ cun 3946 ∅c0 4322 {csn 4628 {cpr 4630 ⟨cop 4634 class class class wbr 5148 ^◡ccnv 5675 Fun wfun 6535 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 1c1 11108 − cmin 11441 ℕcn 12209 2c2 12264 7c7 12269 cdc 12674 ...cfz 13481 ↑cexp 14024 Σcsu 15629 Struct cstr 17076 ndxcnx 17123 Basecbs 17141 distcds 17203 Itvcitv 27674 LineGclng 27675 𝔼cee 28136 Btwn cbtwn 28137 Cgrccgr 28138 EEGceeng 28225

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-seq 13964 df-sum 15630 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-ds 17216 df-itv 27676 df-lng 27677 df-ee 28139 df-cgr 28141 df-eeng 28226

This theorem is referenced by: eengtrkg 28234

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