Theorem isismt 28560

Description: Property of being an isometry. Compare with isismty 37761. (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 3509	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		elex 3509	. . . 4 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V)
3		fveq2 6920	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		isismt.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	f1oeq2d 6858	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→(Base‘ℎ)))
7		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
8		isismt.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘𝐺)
9	7, 8	eqtr4di 2798	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
10	9	oveqd 7465	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
11	10	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏)))
12	5, 11	raleqbidv 3354	. . . . . . . 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 3354	. . . . . . 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 631	. . . . . 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 2811	. . . . 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 6920	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = (Base‘𝐻))
17		isismt.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐻)
18	16, 17	eqtr4di 2798	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = 𝑃)
19	18	f1oeq3d 6859	. . . . . . 7 ⊢ (ℎ = 𝐻 → (𝑓:𝐵–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→𝑃))
20		fveq2 6920	. . . . . . . . . . 11 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = (dist‘𝐻))
21		isismt.m	. . . . . . . . . . 11 ⊢ − = (dist‘𝐻)
22	20, 21	eqtr4di 2798	. . . . . . . . . 10 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = − )
23	22	oveqd 7465	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = ((𝑓‘𝑎) − (𝑓‘𝑏)))
24	23	eqeq1d 2742	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)))
25	24	2ralbidv 3227	. . . . . . 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 631	. . . . . 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 2811	. . . . 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 28559	. . . . 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 7481	. . . . . 6 ⊢ (𝑃 ↑m 𝐵) ∈ V
30		f1of 6862	. . . . . . . . 9 ⊢ (𝑓:𝐵–1-1-onto→𝑃 → 𝑓:𝐵⟶𝑃)
31	17	fvexi 6934	. . . . . . . . . 10 ⊢ 𝑃 ∈ V
32	4	fvexi 6934	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
33	31, 32	elmap 8929	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑃 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝑃)
34	30, 33	sylibr 234	. . . . . . . 8 ⊢ (𝑓:𝐵–1-1-onto→𝑃 → 𝑓 ∈ (𝑃 ↑m 𝐵))
35	34	adantr 480	. . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)) → 𝑓 ∈ (𝑃 ↑m 𝐵))
36	35	abssi 4093	. . . . . 6 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃 ↑m 𝐵)
37	29, 36	ssexi 5340	. . . . 5 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ∈ V
38	15, 27, 28, 37	ovmpo 7610	. . . 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 595	. . 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 2830	. 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 6862	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→𝑃 → 𝐹:𝐵⟶𝑃)
42		fex 7263	. . . . 5 ⊢ ((𝐹:𝐵⟶𝑃 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
43	41, 32, 42	sylancl 585	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝑃 → 𝐹 ∈ V)
44	43	adantr 480	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)) → 𝐹 ∈ V)
45		f1oeq1 6850	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐵–1-1-onto→𝑃 ↔ 𝐹:𝐵–1-1-onto→𝑃))
46		fveq1 6919	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑎) = (𝐹‘𝑎))
47		fveq1 6919	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑏) = (𝐹‘𝑏))
48	46, 47	oveq12d 7466	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑎) − (𝑓‘𝑏)) = ((𝐹‘𝑎) − (𝐹‘𝑏)))
49	48	eqeq1d 2742	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))
50	49	2ralbidv 3227	. . . 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 631	. . 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 3702	. 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 287	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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  Vcvv 3488  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Basecbs 17258  distcds 17320  Ismtcismt 28558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-ismt 28559

This theorem is referenced by:  ismot  28561