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Theorem iscgrg 25606
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 3352 . . . . 5 (𝐺𝑉𝐺 ∈ V)
3 fveq2 6352 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 iscgrg.p . . . . . . . . . . . 12 𝑃 = (Base‘𝐺)
53, 4syl6eqr 2812 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
65oveq1d 6828 . . . . . . . . . 10 (𝑔 = 𝐺 → ((Base‘𝑔) ↑pm ℝ) = (𝑃pm ℝ))
76eleq2d 2825 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ↔ 𝑎 ∈ (𝑃pm ℝ)))
86eleq2d 2825 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑏 ∈ ((Base‘𝑔) ↑pm ℝ) ↔ 𝑏 ∈ (𝑃pm ℝ)))
97, 8anbi12d 749 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ↔ (𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ))))
10 fveq2 6352 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11 iscgrg.m . . . . . . . . . . . . 13 = (dist‘𝐺)
1210, 11syl6eqr 2812 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (dist‘𝑔) = )
1312oveqd 6830 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑎𝑖) (𝑎𝑗)))
1412oveqd 6830 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗)) = ((𝑏𝑖) (𝑏𝑗)))
1513, 14eqeq12d 2775 . . . . . . . . . 10 (𝑔 = 𝐺 → (((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗)) ↔ ((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))
16152ralbidv 3127 . . . . . . . . 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 742 . . . . . . . 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 749 . . . . . . 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 4868 . . . . . 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 25605 . . . . . 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 5272 . . . . . . . 8 ((𝑃pm ℝ) × (𝑃pm ℝ)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ))}
22 ovex 6841 . . . . . . . . 9 (𝑃pm ℝ) ∈ V
2322, 22xpex 7127 . . . . . . . 8 ((𝑃pm ℝ) × (𝑃pm ℝ)) ∈ V
2421, 23eqeltrri 2836 . . . . . . 7 {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ))} ∈ V
25 simpl 474 . . . . . . . 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 5153 . . . . . . 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 4955 . . . . . 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 6444 . . . . 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 2806 . . 3 (𝐺𝑉 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃pm ℝ) ∧ 𝑏 ∈ (𝑃pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗))))})
3130breqd 4815 . 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 5479 . . . . . 6 (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
3332eqeq1d 2762 . . . . 5 (𝑎 = 𝐴 → (dom 𝑎 = dom 𝑏 ↔ dom 𝐴 = dom 𝑏))
3432adantr 472 . . . . . . 7 ((𝑎 = 𝐴𝑖 ∈ dom 𝑎) → dom 𝑎 = dom 𝐴)
35 simpll 807 . . . . . . . . . 10 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → 𝑎 = 𝐴)
3635fveq1d 6354 . . . . . . . . 9 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → (𝑎𝑖) = (𝐴𝑖))
3735fveq1d 6354 . . . . . . . . 9 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → (𝑎𝑗) = (𝐴𝑗))
3836, 37oveq12d 6831 . . . . . . . 8 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → ((𝑎𝑖) (𝑎𝑗)) = ((𝐴𝑖) (𝐴𝑗)))
3938eqeq1d 2762 . . . . . . 7 (((𝑎 = 𝐴𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → (((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗))))
4034, 39raleqbidva 3293 . . . . . 6 ((𝑎 = 𝐴𝑖 ∈ dom 𝑎) → (∀𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ∀𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗))))
4132, 40raleqbidva 3293 . . . . 5 (𝑎 = 𝐴 → (∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖) (𝑎𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗))))
4233, 41anbi12d 749 . . . 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 5479 . . . . . 6 (𝑏 = 𝐵 → dom 𝑏 = dom 𝐵)
4443eqeq2d 2770 . . . . 5 (𝑏 = 𝐵 → (dom 𝐴 = dom 𝑏 ↔ dom 𝐴 = dom 𝐵))
45 fveq1 6351 . . . . . . . 8 (𝑏 = 𝐵 → (𝑏𝑖) = (𝐵𝑖))
46 fveq1 6351 . . . . . . . 8 (𝑏 = 𝐵 → (𝑏𝑗) = (𝐵𝑗))
4745, 46oveq12d 6831 . . . . . . 7 (𝑏 = 𝐵 → ((𝑏𝑖) (𝑏𝑗)) = ((𝐵𝑖) (𝐵𝑗)))
4847eqeq2d 2770 . . . . . 6 (𝑏 = 𝐵 → (((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
49482ralbidv 3127 . . . . 5 (𝑏 = 𝐵 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝑏𝑖) (𝑏𝑗)) ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
5044, 49anbi12d 749 . . . 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 738 . . 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 2760 . . 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 5351 . 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 276 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340   class class class wbr 4804  {copab 4864   × cxp 5264  dom cdm 5266  cfv 6049  (class class class)co 6813  pm cpm 8024  cr 10127  Basecbs 16059  distcds 16152  cgrGccgrg 25604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-cgrg 25605
This theorem is referenced by:  iscgrgd  25607  ercgrg  25611
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