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Mirrors > Home > MPE Home > Th. List > motplusg | Structured version Visualization version GIF version |
Description: The operation for motions is their composition. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
ismot.p | ⊢ 𝑃 = (Base‘𝐺) |
ismot.m | ⊢ − = (dist‘𝐺) |
motgrp.1 | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
motgrp.i | ⊢ 𝐼 = {〈(Base‘ndx), (𝐺Ismt𝐺)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔))〉} |
motplusg.1 | ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) |
motplusg.2 | ⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺)) |
Ref | Expression |
---|---|
motplusg | ⊢ (𝜑 → (𝐹(+g‘𝐼)𝐻) = (𝐹 ∘ 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | motplusg.1 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) | |
2 | motplusg.2 | . 2 ⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺)) | |
3 | coexg 7626 | . . 3 ⊢ ((𝐹 ∈ (𝐺Ismt𝐺) ∧ 𝐻 ∈ (𝐺Ismt𝐺)) → (𝐹 ∘ 𝐻) ∈ V) | |
4 | 1, 2, 3 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ V) |
5 | coeq1 5721 | . . 3 ⊢ (𝑎 = 𝐹 → (𝑎 ∘ 𝑏) = (𝐹 ∘ 𝑏)) | |
6 | coeq2 5722 | . . 3 ⊢ (𝑏 = 𝐻 → (𝐹 ∘ 𝑏) = (𝐹 ∘ 𝐻)) | |
7 | ovex 7181 | . . . . . 6 ⊢ (𝐺Ismt𝐺) ∈ V | |
8 | 7, 7 | mpoex 7769 | . . . . 5 ⊢ (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔)) ∈ V |
9 | motgrp.i | . . . . . 6 ⊢ 𝐼 = {〈(Base‘ndx), (𝐺Ismt𝐺)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔))〉} | |
10 | 9 | grpplusg 16603 | . . . . 5 ⊢ ((𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔)) ∈ V → (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔)) = (+g‘𝐼)) |
11 | 8, 10 | ax-mp 5 | . . . 4 ⊢ (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔)) = (+g‘𝐼) |
12 | coeq1 5721 | . . . . 5 ⊢ (𝑓 = 𝑎 → (𝑓 ∘ 𝑔) = (𝑎 ∘ 𝑔)) | |
13 | coeq2 5722 | . . . . 5 ⊢ (𝑔 = 𝑏 → (𝑎 ∘ 𝑔) = (𝑎 ∘ 𝑏)) | |
14 | 12, 13 | cbvmpov 7241 | . . . 4 ⊢ (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔)) = (𝑎 ∈ (𝐺Ismt𝐺), 𝑏 ∈ (𝐺Ismt𝐺) ↦ (𝑎 ∘ 𝑏)) |
15 | 11, 14 | eqtr3i 2844 | . . 3 ⊢ (+g‘𝐼) = (𝑎 ∈ (𝐺Ismt𝐺), 𝑏 ∈ (𝐺Ismt𝐺) ↦ (𝑎 ∘ 𝑏)) |
16 | 5, 6, 15 | ovmpog 7301 | . 2 ⊢ ((𝐹 ∈ (𝐺Ismt𝐺) ∧ 𝐻 ∈ (𝐺Ismt𝐺) ∧ (𝐹 ∘ 𝐻) ∈ V) → (𝐹(+g‘𝐼)𝐻) = (𝐹 ∘ 𝐻)) |
17 | 1, 2, 4, 16 | syl3anc 1366 | 1 ⊢ (𝜑 → (𝐹(+g‘𝐼)𝐻) = (𝐹 ∘ 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 Vcvv 3493 {cpr 4561 〈cop 4565 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7148 ∈ cmpo 7150 ndxcnx 16472 Basecbs 16475 +gcplusg 16557 distcds 16566 Ismtcismt 26310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 |
This theorem is referenced by: motgrp 26321 |
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