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Theorem motco 26253
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 26251 . . 3 (𝜑𝐹:𝑃1-1-onto𝑃)
6 motco.3 . . . 4 (𝜑𝐻 ∈ (𝐺Ismt𝐺))
71, 2, 3, 6motf1o 26251 . . 3 (𝜑𝐻:𝑃1-1-onto𝑃)
8 f1oco 6630 . . 3 ((𝐹:𝑃1-1-onto𝑃𝐻:𝑃1-1-onto𝑃) → (𝐹𝐻):𝑃1-1-onto𝑃)
95, 7, 8syl2anc 584 . 2 (𝜑 → (𝐹𝐻):𝑃1-1-onto𝑃)
10 f1of 6608 . . . . . . . 8 (𝐻:𝑃1-1-onto𝑃𝐻:𝑃𝑃)
117, 10syl 17 . . . . . . 7 (𝜑𝐻:𝑃𝑃)
1211adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐻:𝑃𝑃)
13 simprl 767 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑎𝑃)
14 fvco3 6753 . . . . . 6 ((𝐻:𝑃𝑃𝑎𝑃) → ((𝐹𝐻)‘𝑎) = (𝐹‘(𝐻𝑎)))
1512, 13, 14syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹𝐻)‘𝑎) = (𝐹‘(𝐻𝑎)))
16 simprr 769 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑏𝑃)
17 fvco3 6753 . . . . . 6 ((𝐻:𝑃𝑃𝑏𝑃) → ((𝐹𝐻)‘𝑏) = (𝐹‘(𝐻𝑏)))
1812, 16, 17syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹𝐻)‘𝑏) = (𝐹‘(𝐻𝑏)))
1915, 18oveq12d 7163 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (((𝐹𝐻)‘𝑎) ((𝐹𝐻)‘𝑏)) = ((𝐹‘(𝐻𝑎)) (𝐹‘(𝐻𝑏))))
203adantr 481 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐺𝑉)
2112, 13ffvelrnd 6844 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐻𝑎) ∈ 𝑃)
2212, 16ffvelrnd 6844 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐻𝑏) ∈ 𝑃)
234adantr 481 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
241, 2, 20, 21, 22, 23motcgr 26249 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐻𝑎)) (𝐹‘(𝐻𝑏))) = ((𝐻𝑎) (𝐻𝑏)))
256adantr 481 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐻 ∈ (𝐺Ismt𝐺))
261, 2, 20, 13, 16, 25motcgr 26249 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐻𝑎) (𝐻𝑏)) = (𝑎 𝑏))
2719, 24, 263eqtrd 2857 . . 3 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (((𝐹𝐻)‘𝑎) ((𝐹𝐻)‘𝑏)) = (𝑎 𝑏))
2827ralrimivva 3188 . 2 (𝜑 → ∀𝑎𝑃𝑏𝑃 (((𝐹𝐻)‘𝑎) ((𝐹𝐻)‘𝑏)) = (𝑎 𝑏))
291, 2ismot 26248 . . 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 709 1 (𝜑 → (𝐹𝐻) ∈ (𝐺Ismt𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  ccom 5552  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  Ismtcismt 26245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  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-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-ismt 26246
This theorem is referenced by:  motgrp  26256
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