Theorem isismt 28765

Description: Property of being an isometry. Compare with isismty 38335. (Contributed by Thierry Arnoux, 13-Dec-2019.)

Hypotheses

Ref	Expression
isismt.b	⊢ 𝐵 = (Base‘𝐺)
isismt.p	⊢ 𝑃 = (Base‘𝐻)
isismt.d	⊢ 𝐷 = (dist‘𝐺)
isismt.m	⊢ − = (dist‘𝐻)

Assertion

Ref	Expression
isismt	⊢ ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏))))

Distinct variable groups:   𝐵,𝑎,𝑏   𝐹,𝑎,𝑏   𝐺,𝑎,𝑏   𝐻,𝑎,𝑏

Allowed substitution hints:   𝐷(𝑎,𝑏)   𝑃(𝑎,𝑏)   − (𝑎,𝑏)   𝑉(𝑎,𝑏)   𝑊(𝑎,𝑏)

Proof of Theorem isismt

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3484	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		elex 3484	. . . 4 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V)
3		fveq2 6879	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		isismt.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2822	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	f1oeq2d 6814	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→(Base‘ℎ)))
7		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
8		isismt.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘𝐺)
9	7, 8	eqtr4di 2822	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
10	9	oveqd 7425	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
11	10	eqeq2d 2780	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏)))
12	5, 11	raleqbidv 3345	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∀𝑏 ∈ (Base‘𝑔)((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏)))
13	5, 12	raleqbidv 3345	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏)))
14	6, 13	anbi12d 643	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏)) ↔ (𝑓:𝐵–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏))))
15	14	abbidv 2835	. . . . 5 ⊢ (𝑔 = 𝐺 → {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏))} = {𝑓 ∣ (𝑓:𝐵–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏))})
16		fveq2 6879	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = (Base‘𝐻))
17		isismt.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐻)
18	16, 17	eqtr4di 2822	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = 𝑃)
19	18	f1oeq3d 6815	. . . . . . 7 ⊢ (ℎ = 𝐻 → (𝑓:𝐵–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→𝑃))
20		fveq2 6879	. . . . . . . . . . 11 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = (dist‘𝐻))
21		isismt.m	. . . . . . . . . . 11 ⊢ − = (dist‘𝐻)
22	20, 21	eqtr4di 2822	. . . . . . . . . 10 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = − )
23	22	oveqd 7425	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = ((𝑓‘𝑎) − (𝑓‘𝑏)))
24	23	eqeq1d 2771	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)))
25	24	2ralbidv 3235	. . . . . . 7 ⊢ (ℎ = 𝐻 → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)))
26	19, 25	anbi12d 643	. . . . . 6 ⊢ (ℎ = 𝐻 → ((𝑓:𝐵–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏)) ↔ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))))
27	26	abbidv 2835	. . . . 5 ⊢ (ℎ = 𝐻 → {𝑓 ∣ (𝑓:𝐵–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏))} = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))})
28		df-ismt 28764	. . . . 5 ⊢ Ismt = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏))})
29		ovex 7441	. . . . . 6 ⊢ (𝑃 ↑m 𝐵) ∈ V
30		f1of 6818	. . . . . . . . 9 ⊢ (𝑓:𝐵–1-1-onto→𝑃 → 𝑓:𝐵⟶𝑃)
31	17	fvexi 6893	. . . . . . . . . 10 ⊢ 𝑃 ∈ V
32	4	fvexi 6893	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
33	31, 32	elmap 8865	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑃 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝑃)
34	30, 33	sylibr 237	. . . . . . . 8 ⊢ (𝑓:𝐵–1-1-onto→𝑃 → 𝑓 ∈ (𝑃 ↑m 𝐵))
35	34	adantr 485	. . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)) → 𝑓 ∈ (𝑃 ↑m 𝐵))
36	35	abssi 4030	. . . . . 6 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃 ↑m 𝐵)
37	29, 36	ssexi 5290	. . . . 5 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ∈ V
38	15, 27, 28, 37	ovmpo 7568	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))})
39	1, 2, 38	syl2an 607	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))})
40	39	eleq2d 2855	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))}))
41		f1of 6818	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→𝑃 → 𝐹:𝐵⟶𝑃)
42		fex 7222	. . . . 5 ⊢ ((𝐹:𝐵⟶𝑃 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
43	41, 32, 42	sylancl 597	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝑃 → 𝐹 ∈ V)
44	43	adantr 485	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)) → 𝐹 ∈ V)
45		f1oeq1 6806	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐵–1-1-onto→𝑃 ↔ 𝐹:𝐵–1-1-onto→𝑃))
46		fveq1 6878	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑎) = (𝐹‘𝑎))
47		fveq1 6878	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑏) = (𝐹‘𝑏))
48	46, 47	oveq12d 7426	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑎) − (𝑓‘𝑏)) = ((𝐹‘𝑎) − (𝐹‘𝑏)))
49	48	eqeq1d 2771	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))
50	49	2ralbidv 3235	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))
51	45, 50	anbi12d 643	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)) ↔ (𝐹:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏))))
52	44, 51	elab3 3654	. 2 ⊢ (𝐹 ∈ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ↔ (𝐹:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))
53	40, 52	bitrdi 290	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  Vcvv 3463  ⟶wf 6530  –1-1-onto→wf1o 6533  ‘cfv 6534  (class class class)co 7408   ↑_m cmap 8820  Basecbs 17265  distcds 17315  Ismtcismt 28763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8822  df-ismt 28764

This theorem is referenced by:  ismot  28766