motcgrg - Metamath Proof Explorer

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Theorem motcgrg 29007

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 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → 𝑇:(0..^𝑛)⟶𝑃)
2	1	adantr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑇:(0..^𝑛)⟶𝑃)
3		simprl 783	. . . . . . . 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 29001	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃)
9		f1of 6824	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑃–1-1-onto→𝑃 → 𝐹:𝑃⟶𝑃)
10	8, 9	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑃⟶𝑃)
11	10	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → 𝐹:𝑃⟶𝑃)
12		fco 6734	. . . . . . . . . . 11 ⊢ ((𝐹:𝑃⟶𝑃 ∧ 𝑇:(0..^𝑛)⟶𝑃) → (𝐹 ∘ 𝑇):(0..^𝑛)⟶𝑃)
13	11, 1, 12	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → (𝐹 ∘ 𝑇):(0..^𝑛)⟶𝑃)
14	13	adantr 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (𝐹 ∘ 𝑇):(0..^𝑛)⟶𝑃)
15	14	fdmd 6720	. . . . . . . 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 2863	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑎 ∈ (0..^𝑛))
17		fvco3 6985	. . . . . . 7 ⊢ ((𝑇:(0..^𝑛)⟶𝑃 ∧ 𝑎 ∈ (0..^𝑛)) → ((𝐹 ∘ 𝑇)‘𝑎) = (𝐹‘(𝑇‘𝑎)))
18	2, 16, 17	syl2anc 596	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → ((𝐹 ∘ 𝑇)‘𝑎) = (𝐹‘(𝑇‘𝑎)))
19		simprr 785	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑏 ∈ dom (𝐹 ∘ 𝑇))
20	19, 15	eleqtrd 2863	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝑏 ∈ (0..^𝑛))
21		fvco3 6985	. . . . . . 7 ⊢ ((𝑇:(0..^𝑛)⟶𝑃 ∧ 𝑏 ∈ (0..^𝑛)) → ((𝐹 ∘ 𝑇)‘𝑏) = (𝐹‘(𝑇‘𝑏)))
22	2, 20, 21	syl2anc 596	. . . . . 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 7438	. . . . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) → 𝐺 ∈ 𝑉)
25	24	adantr 486	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → 𝐺 ∈ 𝑉)
26	2, 16	ffvelcdmd 7085	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (𝑇‘𝑎) ∈ 𝑃)
27	2, 20	ffvelcdmd 7085	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑇:(0..^𝑛)⟶𝑃) ∧ (𝑎 ∈ dom (𝐹 ∘ 𝑇) ∧ 𝑏 ∈ dom (𝐹 ∘ 𝑇))) → (𝑇‘𝑏) ∈ 𝑃)
28	7	ad3antrrr 743	. . . . . 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 28999	. . . . 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 2796	. . . 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 3206	. . 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 13881	. . . . . 6 ⊢ (0..^𝑛) ⊆ ℕ₀
34		nn0ssre 12610	. . . . . 6 ⊢ ℕ₀ ⊆ ℝ
35	33, 34	sstri 3940	. . . . 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 28976	. . 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 14660	. . 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 3171	1 ⊢ (𝜑 → (𝐹 ∘ 𝑇) ∼ 𝑇)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∃wrex 3087 ⊆ wss 3899 {cpr 4586 ⟨cop 4590 class class class wbr 5103 dom cdm 5651 ∘ ccom 5655 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 ℝcr 11199 0cc0 11200 ℕ₀cn0 12606 ..^cfzo 13788 Word cword 14658 ndxcnx 17371 Basecbs 17387 +_gcplusg 17428 distcds 17437 cgrGccgrg 28973 Ismtcismt 28995

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-word 14659 df-cgrg 28974 df-ismt 28996

This theorem is used by: (None)

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