motcgrg - Metamath Proof Explorer

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Theorem motcgrg 26589

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 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → 𝑇:(0..^𝑛)⟶𝑃)
2	1	adantr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑇:(0..^𝑛)⟶𝑃)
3		simprl 771	. . . . . . . 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 26583	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃)
9		f1of 6639	. . . . . . . . . . . . 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 6547	. . . . . . . . . . 11 ⊢ ((𝐹:𝑃⟶𝑃 ∧ 𝑇:(0..^𝑛)⟶𝑃) → (𝐹 ∘ 𝑇):(0..^𝑛)⟶𝑃)
13	11, 1, 12	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → (𝐹 ∘ 𝑇):(0..^𝑛)⟶𝑃)
14	13	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (𝐹 ∘ 𝑇):(0..^𝑛)⟶𝑃)
15	14	fdmd 6534	. . . . . . . 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 2833	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑎 ∈ (0..^𝑛))
17		fvco3 6788	. . . . . . 7 ⊢ ((𝑇:(0..^𝑛)⟶𝑃 ∧ 𝑎 ∈ (0..^𝑛)) → ((𝐹 ∘ 𝑇)‘𝑎) = (𝐹‘(𝑇‘𝑎)))
18	2, 16, 17	syl2anc 587	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → ((𝐹 ∘ 𝑇)‘𝑎) = (𝐹‘(𝑇‘𝑎)))
19		simprr 773	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑏 ∈ dom (𝐹 ∘ 𝑇))
20	19, 15	eleqtrd 2833	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑏 ∈ (0..^𝑛))
21		fvco3 6788	. . . . . . 7 ⊢ ((𝑇:(0..^𝑛)⟶𝑃 ∧ 𝑏 ∈ (0..^𝑛)) → ((𝐹 ∘ 𝑇)‘𝑏) = (𝐹‘(𝑇‘𝑏)))
22	2, 20, 21	syl2anc 587	. . . . . 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 7209	. . . . 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 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝐺 ∈ 𝑉)
26	2, 16	ffvelrnd 6883	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (𝑇‘𝑎) ∈ 𝑃)
27	2, 20	ffvelrnd 6883	. . . . . 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 26581	. . . . 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 2771	. . . 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 3102	. . 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 13288	. . . . . 6 ⊢ (0..^𝑛) ⊆ ℕ₀
34		nn0ssre 12059	. . . . . 6 ⊢ ℕ₀ ⊆ ℝ
35	33, 34	sstri 3896	. . . . 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 26558	. . 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 260	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → (𝐹 ∘ 𝑇) ∼ 𝑇)
39		motcgrg.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ Word 𝑃)
40		iswrd 14036	. . 3 ⊢ (𝑇 ∈ Word 𝑃 ↔ ∃𝑛 ∈ ℕ₀ 𝑇:(0..^𝑛)⟶𝑃)
41	39, 40	sylib 221	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ 𝑇:(0..^𝑛)⟶𝑃)
42	38, 41	r19.29a 3198	1 ⊢ (𝜑 → (𝐹 ∘ 𝑇) ∼ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ⊆ wss 3853 {cpr 4529 ⟨cop 4533 class class class wbr 5039 dom cdm 5536 ∘ ccom 5540 ⟶wf 6354 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 ℝcr 10693 0cc0 10694 ℕ₀cn0 12055 ..^cfzo 13203 Word cword 14034 ndxcnx 16663 Basecbs 16666 +_gcplusg 16749 distcds 16758 cgrGccgrg 26555 Ismtcismt 26577

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-word 14035 df-cgrg 26556 df-ismt 26578

This theorem is referenced by: (None)

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