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Theorem cnvmot 26241
Description: The converse of a motion 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))
Assertion
Ref Expression
cnvmot (𝜑𝐹 ∈ (𝐺Ismt𝐺))

Proof of Theorem cnvmot
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 26238 . . 3 (𝜑𝐹:𝑃1-1-onto𝑃)
6 f1ocnv 6624 . . 3 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃1-1-onto𝑃)
75, 6syl 17 . 2 (𝜑𝐹:𝑃1-1-onto𝑃)
83adantr 481 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐺𝑉)
9 f1of 6612 . . . . . . . 8 (𝐹:𝑃1-1-onto𝑃𝐹:𝑃𝑃)
107, 9syl 17 . . . . . . 7 (𝜑𝐹:𝑃𝑃)
1110adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹:𝑃𝑃)
12 simprl 767 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑎𝑃)
1311, 12ffvelrnd 6848 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑎) ∈ 𝑃)
14 simprr 769 . . . . . 6 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝑏𝑃)
1511, 14ffvelrnd 6848 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹𝑏) ∈ 𝑃)
164adantr 481 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺))
171, 2, 8, 13, 15, 16motcgr 26236 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = ((𝐹𝑎) (𝐹𝑏)))
18 f1ocnvfv2 7028 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑎𝑃) → (𝐹‘(𝐹𝑎)) = 𝑎)
195, 12, 18syl2an2r 681 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑎)) = 𝑎)
20 f1ocnvfv2 7028 . . . . . 6 ((𝐹:𝑃1-1-onto𝑃𝑏𝑃) → (𝐹‘(𝐹𝑏)) = 𝑏)
215, 14, 20syl2an2r 681 . . . . 5 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → (𝐹‘(𝐹𝑏)) = 𝑏)
2219, 21oveq12d 7166 . . . 4 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = (𝑎 𝑏))
2317, 22eqtr3d 2863 . . 3 ((𝜑 ∧ (𝑎𝑃𝑏𝑃)) → ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
2423ralrimivva 3196 . 2 (𝜑 → ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))
251, 2ismot 26235 . . 3 (𝐺𝑉 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))))
263, 25syl 17 . 2 (𝜑 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))))
277, 24, 26mpbir2and 709 1 (𝜑𝐹 ∈ (𝐺Ismt𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  ccnv 5553  wf 6348  1-1-ontowf1o 6351  cfv 6352  (class class class)co 7148  Basecbs 16473  distcds 16564  Ismtcismt 26232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8398  df-ismt 26233
This theorem is referenced by:  motgrp  26243
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