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Theorem iscgrgd 25628
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 6363 . . . . . . 7 (Base‘𝐺) ∈ V
53, 4eqeltri 2835 . . . . . 6 𝑃 ∈ V
6 reex 10239 . . . . . 6 ℝ ∈ V
7 elpm2r 8043 . . . . . 6 (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐴:𝐷𝑃𝐷 ⊆ ℝ)) → 𝐴 ∈ (𝑃pm ℝ))
85, 6, 7mpanl12 720 . . . . 5 ((𝐴:𝐷𝑃𝐷 ⊆ ℝ) → 𝐴 ∈ (𝑃pm ℝ))
91, 2, 8syl2anc 696 . . . 4 (𝜑𝐴 ∈ (𝑃pm ℝ))
10 iscgrgd.b . . . . 5 (𝜑𝐵:𝐷𝑃)
11 elpm2r 8043 . . . . . 6 (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐵:𝐷𝑃𝐷 ⊆ ℝ)) → 𝐵 ∈ (𝑃pm ℝ))
125, 6, 11mpanl12 720 . . . . 5 ((𝐵:𝐷𝑃𝐷 ⊆ ℝ) → 𝐵 ∈ (𝑃pm ℝ))
1310, 2, 12syl2anc 696 . . . 4 (𝜑𝐵 ∈ (𝑃pm ℝ))
149, 13jca 555 . . 3 (𝜑 → (𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)))
1514biantrurd 530 . 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 6212 . . . . 5 (𝐴:𝐷𝑃 → dom 𝐴 = 𝐷)
171, 16syl 17 . . . 4 (𝜑 → dom 𝐴 = 𝐷)
18 fdm 6212 . . . . 5 (𝐵:𝐷𝑃 → dom 𝐵 = 𝐷)
1910, 18syl 17 . . . 4 (𝜑 → dom 𝐵 = 𝐷)
2017, 19eqtr4d 2797 . . 3 (𝜑 → dom 𝐴 = dom 𝐵)
2120biantrurd 530 . 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 25627 . . 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 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715   class class class wbr 4804  dom cdm 5266  wf 6045  cfv 6049  (class class class)co 6814  pm cpm 8026  cr 10147  Basecbs 16079  distcds 16172  cgrGccgrg 25625
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 7115  ax-cnex 10204  ax-resscn 10205
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-ne 2933  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-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-pm 8028  df-cgrg 25626
This theorem is referenced by:  iscgrglt  25629  trgcgrg  25630  motcgrg  25659
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