![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 27477 | . . . . 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 6842 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
12 | 11 | oveq1d 7372 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) − (𝐹‘𝑏)) = ((𝐹‘𝐴) − (𝐹‘𝑏))) |
13 | oveq1 7364 | . . . 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 6842 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵)) | |
16 | 15 | oveq2d 7373 | . . . 4 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝐴) − (𝐹‘𝑏)) = ((𝐹‘𝐴) − (𝐹‘𝐵))) |
17 | oveq2 7365 | . . . 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 3591 | . 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 836 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) − (𝐹‘𝐵)) = (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 distcds 17142 Ismtcismt 27474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-map 8767 df-ismt 27475 |
This theorem is referenced by: motco 27482 cnvmot 27483 motcgrg 27486 motcgr3 27487 |
Copyright terms: Public domain | W3C validator |