motcgrg - Metamath Proof Explorer

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Theorem motcgrg 28489

Description: Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.)

**Hypotheses**
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))⟩}
motcgrg.r	⊢ ∼ = (cgrG‘𝐺)
motcgrg.t	⊢ (𝜑 → 𝑇 ∈ Word 𝑃)
motcgrg.f	⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺))

**Assertion**
Ref	Expression
motcgrg	⊢ (𝜑 → (𝐹 ∘ 𝑇) ∼ 𝑇)

Distinct variable groups: 𝑓,𝐺,𝑔 𝑓,𝐼,𝑔 𝑃,𝑓,𝑔 𝜑,𝑓,𝑔 Allowed substitution hints: ∼ (𝑓,𝑔) 𝑇(𝑓,𝑔) 𝐹(𝑓,𝑔) − (𝑓,𝑔) 𝑉(𝑓,𝑔)

Proof of Theorem motcgrg Dummy variables 𝑎 𝑏 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → 𝑇:(0..^𝑛)⟶𝑃)
2	1	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑇:(0..^𝑛)⟶𝑃)
3		simprl 770	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑎 ∈ dom (𝐹 ∘ 𝑇))
4		ismot.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (Base‘𝐺)
5		ismot.m	. . . . . . . . . . . . . 14 ⊢ − = (dist‘𝐺)
6		motgrp.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
7		motcgrg.f	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺))
8	4, 5, 6, 7	motf1o 28483	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃)
9		f1of 6764	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑃–1-1-onto→𝑃 → 𝐹:𝑃⟶𝑃)
10	8, 9	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑃⟶𝑃)
11	10	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → 𝐹:𝑃⟶𝑃)
12		fco 6676	. . . . . . . . . . 11 ⊢ ((𝐹:𝑃⟶𝑃 ∧ 𝑇:(0..^𝑛)⟶𝑃) → (𝐹 ∘ 𝑇):(0..^𝑛)⟶𝑃)
13	11, 1, 12	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → (𝐹 ∘ 𝑇):(0..^𝑛)⟶𝑃)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (𝐹 ∘ 𝑇):(0..^𝑛)⟶𝑃)
15	14	fdmd 6662	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → dom (𝐹 ∘ 𝑇) = (0..^𝑛))
16	3, 15	eleqtrd 2830	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑎 ∈ (0..^𝑛))
17		fvco3 6922	. . . . . . 7 ⊢ ((𝑇:(0..^𝑛)⟶𝑃 ∧ 𝑎 ∈ (0..^𝑛)) → ((𝐹 ∘ 𝑇)‘𝑎) = (𝐹‘(𝑇‘𝑎)))
18	2, 16, 17	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → ((𝐹 ∘ 𝑇)‘𝑎) = (𝐹‘(𝑇‘𝑎)))
19		simprr 772	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑏 ∈ dom (𝐹 ∘ 𝑇))
20	19, 15	eleqtrd 2830	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑏 ∈ (0..^𝑛))
21		fvco3 6922	. . . . . . 7 ⊢ ((𝑇:(0..^𝑛)⟶𝑃 ∧ 𝑏 ∈ (0..^𝑛)) → ((𝐹 ∘ 𝑇)‘𝑏) = (𝐹‘(𝑇‘𝑏)))
22	2, 20, 21	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → ((𝐹 ∘ 𝑇)‘𝑏) = (𝐹‘(𝑇‘𝑏)))
23	18, 22	oveq12d 7367	. . . . 5 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (((𝐹 ∘ 𝑇)‘𝑎) − ((𝐹 ∘ 𝑇)‘𝑏)) = ((𝐹‘(𝑇‘𝑎)) − (𝐹‘(𝑇‘𝑏))))
24	6	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → 𝐺 ∈ 𝑉)
25	24	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝐺 ∈ 𝑉)
26	2, 16	ffvelcdmd 7019	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (𝑇‘𝑎) ∈ 𝑃)
27	2, 20	ffvelcdmd 7019	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (𝑇‘𝑏) ∈ 𝑃)
28	7	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝐹 ∈ (𝐺Ismt𝐺))
29	4, 5, 25, 26, 27, 28	motcgr 28481	. . . . 5 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → ((𝐹‘(𝑇‘𝑎)) − (𝐹‘(𝑇‘𝑏))) = ((𝑇‘𝑎) − (𝑇‘𝑏)))
30	23, 29	eqtrd 2764	. . . 4 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (((𝐹 ∘ 𝑇)‘𝑎) − ((𝐹 ∘ 𝑇)‘𝑏)) = ((𝑇‘𝑎) − (𝑇‘𝑏)))
31	30	ralrimivva 3172	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → ∀𝑎 ∈ dom (𝐹 ∘ 𝑇)∀𝑏 ∈ dom (𝐹 ∘ 𝑇)(((𝐹 ∘ 𝑇)‘𝑎) − ((𝐹 ∘ 𝑇)‘𝑏)) = ((𝑇‘𝑎) − (𝑇‘𝑏)))
32		motcgrg.r	. . . 4 ⊢ ∼ = (cgrG‘𝐺)
33		fzo0ssnn0 13649	. . . . . 6 ⊢ (0..^𝑛) ⊆ ℕ₀
34		nn0ssre 12388	. . . . . 6 ⊢ ℕ₀ ⊆ ℝ
35	33, 34	sstri 3945	. . . . 5 ⊢ (0..^𝑛) ⊆ ℝ
36	35	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → (0..^𝑛) ⊆ ℝ)
37	4, 5, 32, 24, 36, 13, 1	iscgrgd 28458	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → ((𝐹 ∘ 𝑇) ∼ 𝑇 ↔ ∀𝑎 ∈ dom (𝐹 ∘ 𝑇)∀𝑏 ∈ dom (𝐹 ∘ 𝑇)(((𝐹 ∘ 𝑇)‘𝑎) − ((𝐹 ∘ 𝑇)‘𝑏)) = ((𝑇‘𝑎) − (𝑇‘𝑏))))
38	31, 37	mpbird 257	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → (𝐹 ∘ 𝑇) ∼ 𝑇)
39		motcgrg.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ Word 𝑃)
40		iswrd 14422	. . 3 ⊢ (𝑇 ∈ Word 𝑃 ↔ ∃𝑛 ∈ ℕ₀ 𝑇:(0..^𝑛)⟶𝑃)
41	39, 40	sylib 218	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ 𝑇:(0..^𝑛)⟶𝑃)
42	38, 41	r19.29a 3137	1 ⊢ (𝜑 → (𝐹 ∘ 𝑇) ∼ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ⊆ wss 3903 {cpr 4579 ⟨cop 4583 class class class wbr 5092 dom cdm 5619 ∘ ccom 5623 ⟶wf 6478 –1-1-onto→wf1o 6481 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 ℝcr 11008 0cc0 11009 ℕ₀cn0 12384 ..^cfzo 13557 Word cword 14420 ndxcnx 17104 Basecbs 17120 +_gcplusg 17161 distcds 17170 cgrGccgrg 28455 Ismtcismt 28477

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-n0 12385 df-z 12472 df-uz 12736 df-fz 13411 df-fzo 13558 df-word 14421 df-cgrg 28456 df-ismt 28478

This theorem is referenced by: (None)

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