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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 26329 | . . . . 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 235 | . . 3 ⊢ (𝜑 → (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏))) |
10 | 9 | simprd 499 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)) |
11 | fveq2 6645 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
12 | 11 | oveq1d 7150 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) − (𝐹‘𝑏)) = ((𝐹‘𝐴) − (𝐹‘𝑏))) |
13 | oveq1 7142 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 − 𝑏) = (𝐴 − 𝑏)) | |
14 | 12, 13 | eqeq12d 2814 | . . 3 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏) ↔ ((𝐹‘𝐴) − (𝐹‘𝑏)) = (𝐴 − 𝑏))) |
15 | fveq2 6645 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵)) | |
16 | 15 | oveq2d 7151 | . . . 4 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝐴) − (𝐹‘𝑏)) = ((𝐹‘𝐴) − (𝐹‘𝐵))) |
17 | oveq2 7143 | . . . 4 ⊢ (𝑏 = 𝐵 → (𝐴 − 𝑏) = (𝐴 − 𝐵)) | |
18 | 16, 17 | eqeq12d 2814 | . . 3 ⊢ (𝑏 = 𝐵 → (((𝐹‘𝐴) − (𝐹‘𝑏)) = (𝐴 − 𝑏) ↔ ((𝐹‘𝐴) − (𝐹‘𝐵)) = (𝐴 − 𝐵))) |
19 | 14, 18 | rspc2va 3582 | . 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 Ismtcismt 26326 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-ismt 26327 |
This theorem is referenced by: motco 26334 cnvmot 26335 motcgrg 26338 motcgr3 26339 |
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