Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscgrgd Structured version   Visualization version   GIF version

Theorem iscgrgd 25208
 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 6113 . . . . . . 7 (Base‘𝐺) ∈ V
53, 4eqeltri 2684 . . . . . 6 𝑃 ∈ V
6 reex 9906 . . . . . 6 ℝ ∈ V
7 elpm2r 7761 . . . . . 6 (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐴:𝐷𝑃𝐷 ⊆ ℝ)) → 𝐴 ∈ (𝑃pm ℝ))
85, 6, 7mpanl12 714 . . . . 5 ((𝐴:𝐷𝑃𝐷 ⊆ ℝ) → 𝐴 ∈ (𝑃pm ℝ))
91, 2, 8syl2anc 691 . . . 4 (𝜑𝐴 ∈ (𝑃pm ℝ))
10 iscgrgd.b . . . . 5 (𝜑𝐵:𝐷𝑃)
11 elpm2r 7761 . . . . . 6 (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐵:𝐷𝑃𝐷 ⊆ ℝ)) → 𝐵 ∈ (𝑃pm ℝ))
125, 6, 11mpanl12 714 . . . . 5 ((𝐵:𝐷𝑃𝐷 ⊆ ℝ) → 𝐵 ∈ (𝑃pm ℝ))
1310, 2, 12syl2anc 691 . . . 4 (𝜑𝐵 ∈ (𝑃pm ℝ))
149, 13jca 553 . . 3 (𝜑 → (𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)))
1514biantrurd 528 . 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 5964 . . . . 5 (𝐴:𝐷𝑃 → dom 𝐴 = 𝐷)
171, 16syl 17 . . . 4 (𝜑 → dom 𝐴 = 𝐷)
18 fdm 5964 . . . . 5 (𝐵:𝐷𝑃 → dom 𝐵 = 𝐷)
1910, 18syl 17 . . . 4 (𝜑 → dom 𝐵 = 𝐷)
2017, 19eqtr4d 2647 . . 3 (𝜑 → dom 𝐴 = dom 𝐵)
2120biantrurd 528 . 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 25207 . . 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 300 1 (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑pm cpm 7745  ℝcr 9814  Basecbs 15695  distcds 15777  cgrGccgrg 25205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-cgrg 25206 This theorem is referenced by:  iscgrglt  25209  trgcgrg  25210  motcgrg  25239
 Copyright terms: Public domain W3C validator