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Mirrors > Home > MPE Home > Th. List > idmot | Structured version Visualization version GIF version |
Description: The identity is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
ismot.p | ⊢ 𝑃 = (Base‘𝐺) |
ismot.m | ⊢ − = (dist‘𝐺) |
motgrp.1 | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
Ref | Expression |
---|---|
idmot | ⊢ (𝜑 → ( I ↾ 𝑃) ∈ (𝐺Ismt𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | motgrp.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
2 | f1oi 6475 | . . 3 ⊢ ( I ↾ 𝑃):𝑃–1-1-onto→𝑃 | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ( I ↾ 𝑃):𝑃–1-1-onto→𝑃) |
4 | fvresi 6752 | . . . . 5 ⊢ (𝑎 ∈ 𝑃 → (( I ↾ 𝑃)‘𝑎) = 𝑎) | |
5 | 4 | ad2antrl 715 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (( I ↾ 𝑃)‘𝑎) = 𝑎) |
6 | fvresi 6752 | . . . . 5 ⊢ (𝑏 ∈ 𝑃 → (( I ↾ 𝑃)‘𝑏) = 𝑏) | |
7 | 6 | ad2antll 716 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (( I ↾ 𝑃)‘𝑏) = 𝑏) |
8 | 5, 7 | oveq12d 6988 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((( I ↾ 𝑃)‘𝑎) − (( I ↾ 𝑃)‘𝑏)) = (𝑎 − 𝑏)) |
9 | 8 | ralrimivva 3135 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((( I ↾ 𝑃)‘𝑎) − (( I ↾ 𝑃)‘𝑏)) = (𝑎 − 𝑏)) |
10 | ismot.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
11 | ismot.m | . . . 4 ⊢ − = (dist‘𝐺) | |
12 | 10, 11 | ismot 26013 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (( I ↾ 𝑃) ∈ (𝐺Ismt𝐺) ↔ (( I ↾ 𝑃):𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((( I ↾ 𝑃)‘𝑎) − (( I ↾ 𝑃)‘𝑏)) = (𝑎 − 𝑏)))) |
13 | 12 | biimpar 470 | . 2 ⊢ ((𝐺 ∈ 𝑉 ∧ (( I ↾ 𝑃):𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((( I ↾ 𝑃)‘𝑎) − (( I ↾ 𝑃)‘𝑏)) = (𝑎 − 𝑏))) → ( I ↾ 𝑃) ∈ (𝐺Ismt𝐺)) |
14 | 1, 3, 9, 13 | syl12anc 824 | 1 ⊢ (𝜑 → ( I ↾ 𝑃) ∈ (𝐺Ismt𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∀wral 3082 I cid 5304 ↾ cres 5402 –1-1-onto→wf1o 6181 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 distcds 16420 Ismtcismt 26010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-map 8200 df-ismt 26011 |
This theorem is referenced by: motgrp 26021 |
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