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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 6731 | . . . . . 6 ⊢ 𝑃 ∈ V |
5 | reex 10820 | . . . . . 6 ⊢ ℝ ∈ V | |
6 | elpm2r 8526 | . . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐴 ∈ (𝑃 ↑pm ℝ)) | |
7 | 4, 5, 6 | mpanl12 702 | . . . . 5 ⊢ ((𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐴 ∈ (𝑃 ↑pm ℝ)) |
8 | 1, 2, 7 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑃 ↑pm ℝ)) |
9 | iscgrgd.b | . . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑃) | |
10 | elpm2r 8526 | . . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐵 ∈ (𝑃 ↑pm ℝ)) | |
11 | 4, 5, 10 | mpanl12 702 | . . . . 5 ⊢ ((𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐵 ∈ (𝑃 ↑pm ℝ)) |
12 | 9, 2, 11 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑃 ↑pm ℝ)) |
13 | 8, 12 | jca 515 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ))) |
14 | 13 | biantrurd 536 | . 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 6556 | . . . 4 ⊢ (𝜑 → dom 𝐴 = 𝐷) |
16 | 9 | fdmd 6556 | . . . 4 ⊢ (𝜑 → dom 𝐵 = 𝐷) |
17 | 15, 16 | eqtr4d 2780 | . . 3 ⊢ (𝜑 → dom 𝐴 = dom 𝐵) |
18 | 17 | biantrurd 536 | . 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 26603 | . . 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 315 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ⊆ wss 3866 class class class wbr 5053 dom cdm 5551 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ↑pm cpm 8509 ℝcr 10728 Basecbs 16760 distcds 16811 cgrGccgrg 26601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-pm 8511 df-cgrg 26602 |
This theorem is referenced by: iscgrglt 26605 trgcgrg 26606 motcgrg 26635 |
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