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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 | 3 | fvexi 6677 | . . . . . 6 ⊢ 𝑃 ∈ V |
5 | reex 10616 | . . . . . 6 ⊢ ℝ ∈ V | |
6 | elpm2r 8413 | . . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐴 ∈ (𝑃 ↑pm ℝ)) | |
7 | 4, 5, 6 | mpanl12 698 | . . . . 5 ⊢ ((𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐴 ∈ (𝑃 ↑pm ℝ)) |
8 | 1, 2, 7 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑃 ↑pm ℝ)) |
9 | iscgrgd.b | . . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑃) | |
10 | elpm2r 8413 | . . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐵 ∈ (𝑃 ↑pm ℝ)) | |
11 | 4, 5, 10 | mpanl12 698 | . . . . 5 ⊢ ((𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐵 ∈ (𝑃 ↑pm ℝ)) |
12 | 9, 2, 11 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑃 ↑pm ℝ)) |
13 | 8, 12 | jca 512 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ))) |
14 | 13 | biantrurd 533 | . 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)))))) |
15 | 1 | fdmd 6516 | . . . 4 ⊢ (𝜑 → dom 𝐴 = 𝐷) |
16 | 9 | fdmd 6516 | . . . 4 ⊢ (𝜑 → dom 𝐵 = 𝐷) |
17 | 15, 16 | eqtr4d 2856 | . . 3 ⊢ (𝜑 → dom 𝐴 = dom 𝐵) |
18 | 17 | biantrurd 533 | . 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))))) |
19 | iscgrgd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
20 | iscgrg.m | . . . 4 ⊢ − = (dist‘𝐺) | |
21 | iscgrg.e | . . . 4 ⊢ ∼ = (cgrG‘𝐺) | |
22 | 3, 20, 21 | iscgrg 26225 | . . 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)))))) |
23 | 19, 22 | 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)))))) |
24 | 14, 18, 23 | 3bitr4rd 313 | 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 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 ℝcr 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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