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Theorem iscgrgd 26226
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)
43fvexi 6677 . . . . . 6 𝑃 ∈ V
5 reex 10616 . . . . . 6 ℝ ∈ V
6 elpm2r 8413 . . . . . 6 (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐴:𝐷𝑃𝐷 ⊆ ℝ)) → 𝐴 ∈ (𝑃pm ℝ))
74, 5, 6mpanl12 698 . . . . 5 ((𝐴:𝐷𝑃𝐷 ⊆ ℝ) → 𝐴 ∈ (𝑃pm ℝ))
81, 2, 7syl2anc 584 . . . 4 (𝜑𝐴 ∈ (𝑃pm ℝ))
9 iscgrgd.b . . . . 5 (𝜑𝐵:𝐷𝑃)
10 elpm2r 8413 . . . . . 6 (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐵:𝐷𝑃𝐷 ⊆ ℝ)) → 𝐵 ∈ (𝑃pm ℝ))
114, 5, 10mpanl12 698 . . . . 5 ((𝐵:𝐷𝑃𝐷 ⊆ ℝ) → 𝐵 ∈ (𝑃pm ℝ))
129, 2, 11syl2anc 584 . . . 4 (𝜑𝐵 ∈ (𝑃pm ℝ))
138, 12jca 512 . . 3 (𝜑 → (𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)))
1413biantrurd 533 . 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))))))
151fdmd 6516 . . . 4 (𝜑 → dom 𝐴 = 𝐷)
169fdmd 6516 . . . 4 (𝜑 → dom 𝐵 = 𝐷)
1715, 16eqtr4d 2856 . . 3 (𝜑 → dom 𝐴 = dom 𝐵)
1817biantrurd 533 . 2 (𝜑 → (∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)) ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
19 iscgrgd.g . . 3 (𝜑𝐺𝑉)
20 iscgrg.m . . . 4 = (dist‘𝐺)
21 iscgrg.e . . . 4 = (cgrG‘𝐺)
223, 20, 21iscgrg 26225 . . 3 (𝐺𝑉 → (𝐴 𝐵 ↔ ((𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))))
2319, 22syl 17 . 2 (𝜑 → (𝐴 𝐵 ↔ ((𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))))
2414, 18, 233bitr4rd 313 1 (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933   class class class wbr 5057  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7145  pm cpm 8396  cr 10524  Basecbs 16471  distcds 16562  cgrGccgrg 26223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  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-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-pm 8398  df-cgrg 26224
This theorem is referenced by:  iscgrglt  26227  trgcgrg  26228  motcgrg  26257
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