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Theorem iscgrg 26290
Description: The congruence property for sequences of points. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
iscgrg.p 𝑃 = (Base‘𝐺)
iscgrg.m = (dist‘𝐺)
iscgrg.e = (cgrG‘𝐺)
Assertion
Ref Expression
iscgrg (𝐺𝑉 → (𝐴 𝐵 ↔ ((𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))))
Distinct variable groups:   𝑖,𝑗,𝐺   𝐴,𝑖,𝑗   𝐵,𝑖,𝑗
Allowed substitution hints:   𝑃(𝑖,𝑗)   (𝑖,𝑗)   (𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem iscgrg
Dummy variables 𝑎 𝑏 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscgrg.e . . . 4 = (cgrG‘𝐺)
2 elex 3511 . . . . 5 (𝐺𝑉𝐺 ∈ V)
3 fveq2 6663 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 iscgrg.p . . . . . . . . . . . 12 𝑃 = (Base‘𝐺)
53, 4syl6eqr 2872 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
65oveq1d 7163 . . . . . . . . . 10 (𝑔 = 𝐺 → ((Base‘𝑔) ↑pm ℝ) = (𝑃pm ℝ))
76eleq2d 2896 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ↔ 𝑎 ∈ (𝑃pm ℝ)))
86eleq2d 2896 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑏 ∈ ((Base‘𝑔) ↑pm ℝ) ↔ 𝑏 ∈ (𝑃pm ℝ)))
97, 8anbi12d 632 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ↔ (𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ))))
10 fveq2 6663 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11 iscgrg.m . . . . . . . . . . . . 13 = (dist‘𝐺)
1210, 11syl6eqr 2872 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (dist‘𝑔) = )
1312oveqd 7165 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑎𝑖) (𝑎𝑗)))
1412oveqd 7165 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗)) = ((𝑏𝑖) (𝑏𝑗)))
1513, 14eqeq12d 2835 . . . . . . . . . 10 (𝑔 = 𝐺 → (((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗)) ↔ ((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))
16152ralbidv 3197 . . . . . . . . 9 (𝑔 = 𝐺 → (∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗)) ↔ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))
1716anbi2d 630 . . . . . . . 8 (𝑔 = 𝐺 → ((dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))) ↔ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗)))))
189, 17anbi12d 632 . . . . . . 7 (𝑔 = 𝐺 → (((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗)))) ↔ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))))
1918opabbidv 5123 . . . . . 6 (𝑔 = 𝐺 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))})
20 df-cgrg 26289 . . . . . 6 cgrG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))))})
21 df-xp 5554 . . . . . . . 8 ((𝑃pm ℝ) × (𝑃pm ℝ)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ))}
22 ovex 7181 . . . . . . . . 9 (𝑃pm ℝ) ∈ V
2322, 22xpex 7468 . . . . . . . 8 ((𝑃pm ℝ) × (𝑃pm ℝ)) ∈ V
2421, 23eqeltrri 2908 . . . . . . 7 {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ))} ∈ V
25 simpl 485 . . . . . . . 8 (((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗)))) → (𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)))
2625ssopab2i 5428 . . . . . . 7 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))} ⊆ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ))}
2724, 26ssexi 5217 . . . . . 6 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))} ∈ V
2819, 20, 27fvmpt 6761 . . . . 5 (𝐺 ∈ V → (cgrG‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))})
292, 28syl 17 . . . 4 (𝐺𝑉 → (cgrG‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))})
301, 29syl5eq 2866 . . 3 (𝐺𝑉 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))})
3130breqd 5068 . 2 (𝐺𝑉 → (𝐴 𝐵𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))}𝐵))
32 dmeq 5765 . . . . . 6 (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
3332eqeq1d 2821 . . . . 5 (𝑎 = 𝐴 → (dom 𝑎 = dom 𝑏 ↔ dom 𝐴 = dom 𝑏))
3432adantr 483 . . . . . . 7 ((𝑎 = 𝐴𝑖 ∈ dom 𝑎) → dom 𝑎 = dom 𝐴)
35 simpll 765 . . . . . . . . . 10 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → 𝑎 = 𝐴)
3635fveq1d 6665 . . . . . . . . 9 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → (𝑎𝑖) = (𝐴𝑖))
3735fveq1d 6665 . . . . . . . . 9 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → (𝑎𝑗) = (𝐴𝑗))
3836, 37oveq12d 7166 . . . . . . . 8 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → ((𝑎𝑖) (𝑎𝑗)) = ((𝐴𝑖) (𝐴𝑗)))
3938eqeq1d 2821 . . . . . . 7 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → (((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗))))
4034, 39raleqbidva 3424 . . . . . 6 ((𝑎 = 𝐴𝑖 ∈ dom 𝑎) → (∀𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ∀𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗))))
4132, 40raleqbidva 3424 . . . . 5 (𝑎 = 𝐴 → (∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗))))
4233, 41anbi12d 632 . . . 4 (𝑎 = 𝐴 → ((dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))) ↔ (dom 𝐴 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗)))))
43 dmeq 5765 . . . . . 6 (𝑏 = 𝐵 → dom 𝑏 = dom 𝐵)
4443eqeq2d 2830 . . . . 5 (𝑏 = 𝐵 → (dom 𝐴 = dom 𝑏 ↔ dom 𝐴 = dom 𝐵))
45 fveq1 6662 . . . . . . . 8 (𝑏 = 𝐵 → (𝑏𝑖) = (𝐵𝑖))
46 fveq1 6662 . . . . . . . 8 (𝑏 = 𝐵 → (𝑏𝑗) = (𝐵𝑗))
4745, 46oveq12d 7166 . . . . . . 7 (𝑏 = 𝐵 → ((𝑏𝑖) (𝑏𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
4847eqeq2d 2830 . . . . . 6 (𝑏 = 𝐵 → (((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
49482ralbidv 3197 . . . . 5 (𝑏 = 𝐵 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
5044, 49anbi12d 632 . . . 4 (𝑏 = 𝐵 → ((dom 𝐴 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗))) ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
5142, 50sylan9bb 512 . . 3 ((𝑎 = 𝐴𝑏 = 𝐵) → ((dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))) ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
52 eqid 2819 . . 3 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))}
5351, 52brab2a 5637 . 2 (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))}𝐵 ↔ ((𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
5431, 53syl6bb 289 1 (𝐺𝑉 → (𝐴 𝐵 ↔ ((𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wral 3136  Vcvv 3493   class class class wbr 5057  {copab 5119   × cxp 5546  dom cdm 5548  cfv 6348  (class class class)co 7148  pm cpm 8399  cr 10528  Basecbs 16475  distcds 16566  cgrGccgrg 26288
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-cgrg 26289
This theorem is referenced by:  iscgrgd  26291  ercgrg  26295
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