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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 25230 | . . . . 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 221 | . . 3 ⊢ (𝜑 → (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏))) |
10 | 9 | simprd 478 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)) |
11 | fveq2 6103 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴)) | |
12 | 11 | oveq1d 6564 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) − (𝐹‘𝑏)) = ((𝐹‘𝐴) − (𝐹‘𝑏))) |
13 | oveq1 6556 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 − 𝑏) = (𝐴 − 𝑏)) | |
14 | 12, 13 | eqeq12d 2625 | . . 3 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏) ↔ ((𝐹‘𝐴) − (𝐹‘𝑏)) = (𝐴 − 𝑏))) |
15 | fveq2 6103 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵)) | |
16 | 15 | oveq2d 6565 | . . . 4 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝐴) − (𝐹‘𝑏)) = ((𝐹‘𝐴) − (𝐹‘𝐵))) |
17 | oveq2 6557 | . . . 4 ⊢ (𝑏 = 𝐵 → (𝐴 − 𝑏) = (𝐴 − 𝐵)) | |
18 | 16, 17 | eqeq12d 2625 | . . 3 ⊢ (𝑏 = 𝐵 → (((𝐹‘𝐴) − (𝐹‘𝑏)) = (𝐴 − 𝑏) ↔ ((𝐹‘𝐴) − (𝐹‘𝐵)) = (𝐴 − 𝐵))) |
19 | 14, 18 | rspc2va 3294 | . 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 1317 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) − (𝐹‘𝐵)) = (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 Ismtcismt 25227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-ismt 25228 |
This theorem is referenced by: motco 25235 cnvmot 25236 motcgrg 25239 motcgr3 25240 |
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