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Theorem iscgrgd 25308
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 (𝜑𝐺𝑉)
iscgrgd.d (𝜑𝐷 ⊆ ℝ)
iscgrgd.a (𝜑𝐴:𝐷𝑃)
iscgrgd.b (𝜑𝐵:𝐷𝑃)
Assertion
Ref Expression
iscgrgd (𝜑 → (𝐴 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:   𝜑(𝑖,𝑗)   𝐷(𝑖,𝑗)   𝑃(𝑖,𝑗)   (𝑖,𝑗)   (𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem iscgrgd
StepHypRef Expression
1 iscgrgd.a . . . . 5 (𝜑𝐴:𝐷𝑃)
2 iscgrgd.d . . . . 5 (𝜑𝐷 ⊆ ℝ)
3 iscgrg.p . . . . . . 7 𝑃 = (Base‘𝐺)
4 fvex 6158 . . . . . . 7 (Base‘𝐺) ∈ V
53, 4eqeltri 2694 . . . . . 6 𝑃 ∈ V
6 reex 9971 . . . . . 6 ℝ ∈ V
7 elpm2r 7819 . . . . . 6 (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐴:𝐷𝑃𝐷 ⊆ ℝ)) → 𝐴 ∈ (𝑃pm ℝ))
85, 6, 7mpanl12 717 . . . . 5 ((𝐴:𝐷𝑃𝐷 ⊆ ℝ) → 𝐴 ∈ (𝑃pm ℝ))
91, 2, 8syl2anc 692 . . . 4 (𝜑𝐴 ∈ (𝑃pm ℝ))
10 iscgrgd.b . . . . 5 (𝜑𝐵:𝐷𝑃)
11 elpm2r 7819 . . . . . 6 (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐵:𝐷𝑃𝐷 ⊆ ℝ)) → 𝐵 ∈ (𝑃pm ℝ))
125, 6, 11mpanl12 717 . . . . 5 ((𝐵:𝐷𝑃𝐷 ⊆ ℝ) → 𝐵 ∈ (𝑃pm ℝ))
1310, 2, 12syl2anc 692 . . . 4 (𝜑𝐵 ∈ (𝑃pm ℝ))
149, 13jca 554 . . 3 (𝜑 → (𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)))
1514biantrurd 529 . 2 (𝜑 → ((dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))) ↔ ((𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))))
16 fdm 6008 . . . . 5 (𝐴:𝐷𝑃 → dom 𝐴 = 𝐷)
171, 16syl 17 . . . 4 (𝜑 → dom 𝐴 = 𝐷)
18 fdm 6008 . . . . 5 (𝐵:𝐷𝑃 → dom 𝐵 = 𝐷)
1910, 18syl 17 . . . 4 (𝜑 → dom 𝐵 = 𝐷)
2017, 19eqtr4d 2658 . . 3 (𝜑 → dom 𝐴 = dom 𝐵)
2120biantrurd 529 . 2 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
22 iscgrgd.g . . 3 (𝜑𝐺𝑉)
23 iscgrg.m . . . 4 = (dist‘𝐺)
24 iscgrg.e . . . 4 = (cgrG‘𝐺)
253, 23, 24iscgrg 25307 . . 3 (𝐺𝑉 → (𝐴 𝐵 ↔ ((𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))))
2622, 25syl 17 . 2 (𝜑 → (𝐴 𝐵 ↔ ((𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))))
2715, 21, 263bitr4rd 301 1 (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555   class class class wbr 4613  dom cdm 5074  wf 5843  cfv 5847  (class class class)co 6604  pm cpm 7803  cr 9879  Basecbs 15781  distcds 15871  cgrGccgrg 25305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-pm 7805  df-cgrg 25306
This theorem is referenced by:  iscgrglt  25309  trgcgrg  25310  motcgrg  25339
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