Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  motco Structured version   Visualization version   GIF version

Theorem motco 26341
 Description: The composition of two motions is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Hypotheses
Ref Expression
ismot.p 𝑃 = (Base‘𝐺)
ismot.m = (dist‘𝐺)
motgrp.1 (𝜑𝐺𝑉)
motco.2 (𝜑𝐹 ∈ (𝐺Ismt𝐺))
motco.3 (𝜑𝐻 ∈ (𝐺Ismt𝐺))
Assertion
Ref Expression
motco (𝜑 → (𝐹𝐻) ∈ (𝐺Ismt𝐺))

Proof of Theorem motco
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismot.p . . . 4 𝑃 = (Base‘𝐺)
2 ismot.m . . . 4 = (dist‘𝐺)
3 motgrp.1 . . . 4 (𝜑𝐺𝑉)
4 motco.2 . . . 4 (𝜑𝐹 ∈ (𝐺Ismt𝐺))
51, 2, 3, 4motf1o 26339 . . 3 (𝜑𝐹:𝑃1-1-onto𝑃)
6 motco.3 . . . 4 (𝜑𝐻 ∈ (𝐺Ismt𝐺))
71, 2, 3, 6motf1o 26339 . . 3 (𝜑𝐻:𝑃1-1-onto𝑃)
8 f1oco 6612 . . 3 ((𝐹:𝑃1-1-onto𝑃𝐻:𝑃1-1-onto𝑃) → (𝐹𝐻):𝑃1-1-onto𝑃)
95, 7, 8syl2anc 587 . 2 (𝜑 → (𝐹𝐻):𝑃1-1-onto𝑃)
10 f1of 6590 . . . . . . . 8 (𝐻:𝑃1-1-onto𝑃𝐻:𝑃𝑃)
117, 10syl 17 . . . . . . 7 (𝜑𝐻:𝑃𝑃)
1211adantr 484 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐻:𝑃𝑃)
13 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑎𝑃)
14 fvco3 6737 . . . . . 6 ((𝐻:𝑃𝑃𝑎𝑃) → ((𝐹𝐻)‘𝑎) = (𝐹‘(𝐻𝑎)))
1512, 13, 14syl2anc 587 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹𝐻)‘𝑎) = (𝐹‘(𝐻𝑎)))
16 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑏𝑃)
17 fvco3 6737 . . . . . 6 ((𝐻:𝑃𝑃𝑏𝑃) → ((𝐹𝐻)‘𝑏) = (𝐹‘(𝐻𝑏)))
1812, 16, 17syl2anc 587 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹𝐻)‘𝑏) = (𝐹‘(𝐻𝑏)))
1915, 18oveq12d 7153 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (((𝐹𝐻)‘𝑎) ((𝐹𝐻)‘𝑏)) = ((𝐹‘(𝐻𝑎)) (𝐹‘(𝐻𝑏))))
203adantr 484 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐺𝑉)
2112, 13ffvelrnd 6829 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐻𝑎) ∈ 𝑃)
2212, 16ffvelrnd 6829 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐻𝑏) ∈ 𝑃)
234adantr 484 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
241, 2, 20, 21, 22, 23motcgr 26337 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐻𝑎)) (𝐹‘(𝐻𝑏))) = ((𝐻𝑎) (𝐻𝑏)))
256adantr 484 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐻 ∈ (𝐺Ismt𝐺))
261, 2, 20, 13, 16, 25motcgr 26337 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐻𝑎) (𝐻𝑏)) = (𝑎 𝑏))
2719, 24, 263eqtrd 2837 . . 3 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (((𝐹𝐻)‘𝑎) ((𝐹𝐻)‘𝑏)) = (𝑎 𝑏))
2827ralrimivva 3156 . 2 (𝜑 → ∀𝑎𝑃𝑏𝑃 (((𝐹𝐻)‘𝑎) ((𝐹𝐻)‘𝑏)) = (𝑎 𝑏))
291, 2ismot 26336 . . 3 (𝐺𝑉 → ((𝐹𝐻) ∈ (𝐺Ismt𝐺) ↔ ((𝐹𝐻):𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 (((𝐹𝐻)‘𝑎) ((𝐹𝐻)‘𝑏)) = (𝑎 𝑏))))
303, 29syl 17 . 2 (𝜑 → ((𝐹𝐻) ∈ (𝐺Ismt𝐺) ↔ ((𝐹𝐻):𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 (((𝐹𝐻)‘𝑎) ((𝐹𝐻)‘𝑏)) = (𝑎 𝑏))))
319, 28, 30mpbir2and 712 1 (𝜑 → (𝐹𝐻) ∈ (𝐺Ismt𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∘ ccom 5523  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  distcds 16568  Ismtcismt 26333 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 7443 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 8393  df-ismt 26334 This theorem is referenced by:  motgrp  26344
 Copyright terms: Public domain W3C validator