Theorem iscgrgd 25408

Description: The property for two sequences 𝐴 and 𝐵 of points to be congruent. (Contributed by Thierry Arnoux, 3-Apr-2019.)

Hypotheses

Ref	Expression
iscgrg.p	⊢ 𝑃 = (Base‘𝐺)
iscgrg.m	⊢ − = (dist‘𝐺)
iscgrg.e	⊢ ∼ = (cgrG‘𝐺)
iscgrgd.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
iscgrgd.d	⊢ (𝜑 → 𝐷 ⊆ ℝ)
iscgrgd.a	⊢ (𝜑 → 𝐴:𝐷⟶𝑃)
iscgrgd.b	⊢ (𝜑 → 𝐵:𝐷⟶𝑃)

Assertion

Ref	Expression
iscgrgd	⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))

Distinct variable groups:   𝑖,𝑗,𝐺   𝐴,𝑖,𝑗   𝐵,𝑖,𝑗

Allowed substitution hints:   𝜑(𝑖,𝑗)   𝐷(𝑖,𝑗)   𝑃(𝑖,𝑗)   ∼ (𝑖,𝑗)   − (𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem iscgrgd

Step	Hyp	Ref	Expression
1		iscgrgd.a	. . . . 5 ⊢ (𝜑 → 𝐴:𝐷⟶𝑃)
2		iscgrgd.d	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
3		iscgrg.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
4		fvex 6201	. . . . . . 7 ⊢ (Base‘𝐺) ∈ V
5	3, 4	eqeltri 2697	. . . . . 6 ⊢ 𝑃 ∈ V
6		reex 10027	. . . . . 6 ⊢ ℝ ∈ V
7		elpm2r 7875	. . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐴 ∈ (𝑃 ↑pm ℝ))
8	5, 6, 7	mpanl12 718	. . . . 5 ⊢ ((𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐴 ∈ (𝑃 ↑pm ℝ))
9	1, 2, 8	syl2anc 693	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑃 ↑pm ℝ))
10		iscgrgd.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑃)
11		elpm2r 7875	. . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐵 ∈ (𝑃 ↑pm ℝ))
12	5, 6, 11	mpanl12 718	. . . . 5 ⊢ ((𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐵 ∈ (𝑃 ↑pm ℝ))
13	10, 2, 12	syl2anc 693	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑃 ↑pm ℝ))
14	9, 13	jca 554	. . 3 ⊢ (𝜑 → (𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ)))
15	14	biantrurd 529	. 2 ⊢ (𝜑 → ((dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))) ↔ ((𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))))
16		fdm 6051	. . . . 5 ⊢ (𝐴:𝐷⟶𝑃 → dom 𝐴 = 𝐷)
17	1, 16	syl 17	. . . 4 ⊢ (𝜑 → dom 𝐴 = 𝐷)
18		fdm 6051	. . . . 5 ⊢ (𝐵:𝐷⟶𝑃 → dom 𝐵 = 𝐷)
19	10, 18	syl 17	. . . 4 ⊢ (𝜑 → dom 𝐵 = 𝐷)
20	17, 19	eqtr4d 2659	. . 3 ⊢ (𝜑 → dom 𝐴 = dom 𝐵)
21	20	biantrurd 529	. 2 ⊢ (𝜑 → (∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)) ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))
22		iscgrgd.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
23		iscgrg.m	. . . 4 ⊢ − = (dist‘𝐺)
24		iscgrg.e	. . . 4 ⊢ ∼ = (cgrG‘𝐺)
25	3, 23, 24	iscgrg 25407	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))))
26	22, 25	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))))
27	15, 21, 26	3bitr4rd 301	1 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_pm cpm 7858  ℝcr 9935  Basecbs 15857  distcds 15950  cgrGccgrg 25405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-cgrg 25406

This theorem is referenced by:  iscgrglt  25409  trgcgrg  25410  motcgrg  25439