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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 6684 | . . . . . 6 ⊢ 𝑃 ∈ V |
5 | reex 10628 | . . . . . 6 ⊢ ℝ ∈ V | |
6 | elpm2r 8424 | . . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐴 ∈ (𝑃 ↑pm ℝ)) | |
7 | 4, 5, 6 | mpanl12 700 | . . . . 5 ⊢ ((𝐴:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐴 ∈ (𝑃 ↑pm ℝ)) |
8 | 1, 2, 7 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑃 ↑pm ℝ)) |
9 | iscgrgd.b | . . . . 5 ⊢ (𝜑 → 𝐵:𝐷⟶𝑃) | |
10 | elpm2r 8424 | . . . . . 6 ⊢ (((𝑃 ∈ V ∧ ℝ ∈ V) ∧ (𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ)) → 𝐵 ∈ (𝑃 ↑pm ℝ)) | |
11 | 4, 5, 10 | mpanl12 700 | . . . . 5 ⊢ ((𝐵:𝐷⟶𝑃 ∧ 𝐷 ⊆ ℝ) → 𝐵 ∈ (𝑃 ↑pm ℝ)) |
12 | 9, 2, 11 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑃 ↑pm ℝ)) |
13 | 8, 12 | jca 514 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ))) |
14 | 13 | biantrurd 535 | . 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 6523 | . . . 4 ⊢ (𝜑 → dom 𝐴 = 𝐷) |
16 | 9 | fdmd 6523 | . . . 4 ⊢ (𝜑 → dom 𝐵 = 𝐷) |
17 | 15, 16 | eqtr4d 2859 | . . 3 ⊢ (𝜑 → dom 𝐴 = dom 𝐵) |
18 | 17 | biantrurd 535 | . 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 26298 | . . 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 314 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑pm cpm 8407 ℝcr 10536 Basecbs 16483 distcds 16574 cgrGccgrg 26296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-pm 8409 df-cgrg 26297 |
This theorem is referenced by: iscgrglt 26300 trgcgrg 26301 motcgrg 26330 |
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