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Mirrors > Home > MPE Home > Th. List > iscgrgd | Structured version Visualization version GIF version |
Description: The property for two sequences 𝐴 and 𝐵 of points to be congruent. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
Ref | Expression |
---|---|
iscgrg.p | ⊢ 𝑃 = (Base‘𝐺) |
iscgrg.m | ⊢ − = (dist‘𝐺) |
iscgrg.e | ⊢ ∼ = (cgrG‘𝐺) |
iscgrgd.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
iscgrgd.d | ⊢ (𝜑 → 𝐷 ⊆ ℝ) |
iscgrgd.a | ⊢ (𝜑 → 𝐴:𝐷⟶𝑃) |
iscgrgd.b | ⊢ (𝜑 → 𝐵:𝐷⟶𝑃) |
Ref | Expression |
---|---|
iscgrgd | ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscgrgd.a | . . . . 5 ⊢ (𝜑 → 𝐴:𝐷⟶𝑃) | |
2 | iscgrgd.d | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ℝ) | |
3 | iscgrg.p | . . . . . . 7 ⊢ 𝑃 = (Base‘𝐺) | |
4 | fvex 6113 | . . . . . . 7 ⊢ (Base‘𝐺) ∈ V | |
5 | 3, 4 | eqeltri 2684 | . . . . . 6 ⊢ 𝑃 ∈ V |
6 | reex 9906 | . . . . . 6 ⊢ ℝ ∈ V | |
7 | elpm2r 7761 | . . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐴 ∈ (𝑃 ↑pm ℝ)) | |
8 | 5, 6, 7 | mpanl12 714 | . . . . 5 ⊢ ((𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐴 ∈ (𝑃 ↑pm ℝ)) |
9 | 1, 2, 8 | syl2anc 691 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑃 ↑pm ℝ)) |
10 | iscgrgd.b | . . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑃) | |
11 | elpm2r 7761 | . . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐵 ∈ (𝑃 ↑pm ℝ)) | |
12 | 5, 6, 11 | mpanl12 714 | . . . . 5 ⊢ ((𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐵 ∈ (𝑃 ↑pm ℝ)) |
13 | 10, 2, 12 | syl2anc 691 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑃 ↑pm ℝ)) |
14 | 9, 13 | jca 553 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ))) |
15 | 14 | biantrurd 528 | . 2 ⊢ (𝜑 → ((dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))) ↔ ((𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))) |
16 | fdm 5964 | . . . . 5 ⊢ (𝐴:𝐷⟶𝑃 → dom 𝐴 = 𝐷) | |
17 | 1, 16 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐴 = 𝐷) |
18 | fdm 5964 | . . . . 5 ⊢ (𝐵:𝐷⟶𝑃 → dom 𝐵 = 𝐷) | |
19 | 10, 18 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐵 = 𝐷) |
20 | 17, 19 | eqtr4d 2647 | . . 3 ⊢ (𝜑 → dom 𝐴 = dom 𝐵) |
21 | 20 | biantrurd 528 | . 2 ⊢ (𝜑 → (∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)) ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))) |
22 | iscgrgd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
23 | iscgrg.m | . . . 4 ⊢ − = (dist‘𝐺) | |
24 | iscgrg.e | . . . 4 ⊢ ∼ = (cgrG‘𝐺) | |
25 | 3, 23, 24 | iscgrg 25207 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))) |
26 | 22, 25 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))) |
27 | 15, 21, 26 | 3bitr4rd 300 | 1 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))) |
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 |
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