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Mirrors > Home > MPE Home > Th. List > motcgr | Structured version Visualization version GIF version |
Description: Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
ismot.p | ⊢ 𝑃 = (Base‘𝐺) |
ismot.m | ⊢ − = (dist‘𝐺) |
motgrp.1 | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
motcgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
motcgr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
motcgr.f | ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) |
Ref | Expression |
---|---|
motcgr | ⊢ (𝜑 → ((𝐹‘𝐴) − (𝐹‘𝐵)) = (𝐴 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | motcgr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
2 | motcgr.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
3 | motcgr.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) | |
4 | motgrp.1 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
5 | ismot.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
6 | ismot.m | . . . . . 6 ⊢ − = (dist‘𝐺) | |
7 | 5, 6 | ismot 27186 | . . . . 5 ⊢ (𝐺 ∈ 𝑉 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)))) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)))) |
9 | 3, 8 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏))) |
10 | 9 | simprd 496 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)) |
11 | fveq2 6826 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
12 | 11 | oveq1d 7353 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) − (𝐹‘𝑏)) = ((𝐹‘𝐴) − (𝐹‘𝑏))) |
13 | oveq1 7345 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 − 𝑏) = (𝐴 − 𝑏)) | |
14 | 12, 13 | eqeq12d 2752 | . . 3 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏) ↔ ((𝐹‘𝐴) − (𝐹‘𝑏)) = (𝐴 − 𝑏))) |
15 | fveq2 6826 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵)) | |
16 | 15 | oveq2d 7354 | . . . 4 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝐴) − (𝐹‘𝑏)) = ((𝐹‘𝐴) − (𝐹‘𝐵))) |
17 | oveq2 7346 | . . . 4 ⊢ (𝑏 = 𝐵 → (𝐴 − 𝑏) = (𝐴 − 𝐵)) | |
18 | 16, 17 | eqeq12d 2752 | . . 3 ⊢ (𝑏 = 𝐵 → (((𝐹‘𝐴) − (𝐹‘𝑏)) = (𝐴 − 𝑏) ↔ ((𝐹‘𝐴) − (𝐹‘𝐵)) = (𝐴 − 𝐵))) |
19 | 14, 18 | rspc2va 3580 | . 2 ⊢ (((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)) → ((𝐹‘𝐴) − (𝐹‘𝐵)) = (𝐴 − 𝐵)) |
20 | 1, 2, 10, 19 | syl21anc 835 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) − (𝐹‘𝐵)) = (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 –1-1-onto→wf1o 6479 ‘cfv 6480 (class class class)co 7338 Basecbs 17010 distcds 17069 Ismtcismt 27183 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-id 5519 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-ov 7341 df-oprab 7342 df-mpo 7343 df-map 8689 df-ismt 27184 |
This theorem is referenced by: motco 27191 cnvmot 27192 motcgrg 27195 motcgr3 27196 |
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