Theorem isismt 28606

Description: Property of being an isometry. Compare with isismty 38002. (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 3461	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		elex 3461	. . . 4 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V)
3		fveq2 6834	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		isismt.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	f1oeq2d 6770	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→(Base‘ℎ)))
7		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
8		isismt.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘𝐺)
9	7, 8	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
10	9	oveqd 7375	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
11	10	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏)))
12	5, 11	raleqbidv 3316	. . . . . . . 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 3316	. . . . . . 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 632	. . . . . 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 2802	. . . . 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 6834	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = (Base‘𝐻))
17		isismt.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐻)
18	16, 17	eqtr4di 2789	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = 𝑃)
19	18	f1oeq3d 6771	. . . . . . 7 ⊢ (ℎ = 𝐻 → (𝑓:𝐵–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→𝑃))
20		fveq2 6834	. . . . . . . . . . 11 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = (dist‘𝐻))
21		isismt.m	. . . . . . . . . . 11 ⊢ − = (dist‘𝐻)
22	20, 21	eqtr4di 2789	. . . . . . . . . 10 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = − )
23	22	oveqd 7375	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = ((𝑓‘𝑎) − (𝑓‘𝑏)))
24	23	eqeq1d 2738	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)))
25	24	2ralbidv 3200	. . . . . . 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 632	. . . . . 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 2802	. . . . 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 28605	. . . . 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 7391	. . . . . 6 ⊢ (𝑃 ↑m 𝐵) ∈ V
30		f1of 6774	. . . . . . . . 9 ⊢ (𝑓:𝐵–1-1-onto→𝑃 → 𝑓:𝐵⟶𝑃)
31	17	fvexi 6848	. . . . . . . . . 10 ⊢ 𝑃 ∈ V
32	4	fvexi 6848	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
33	31, 32	elmap 8809	. . . . . . . . 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 4020	. . . . . 6 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃 ↑m 𝐵)
37	29, 36	ssexi 5267	. . . . 5 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ∈ V
38	15, 27, 28, 37	ovmpo 7518	. . . 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 596	. . 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 2822	. 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 6774	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→𝑃 → 𝐹:𝐵⟶𝑃)
42		fex 7172	. . . . 5 ⊢ ((𝐹:𝐵⟶𝑃 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
43	41, 32, 42	sylancl 586	. . . 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 6762	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐵–1-1-onto→𝑃 ↔ 𝐹:𝐵–1-1-onto→𝑃))
46		fveq1 6833	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑎) = (𝐹‘𝑎))
47		fveq1 6833	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑏) = (𝐹‘𝑏))
48	46, 47	oveq12d 7376	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑎) − (𝑓‘𝑏)) = ((𝐹‘𝑎) − (𝐹‘𝑏)))
49	48	eqeq1d 2738	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))
50	49	2ralbidv 3200	. . . 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 632	. . 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 3641	. 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 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  Vcvv 3440  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  Basecbs 17136  distcds 17186  Ismtcismt 28604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-ismt 28605

This theorem is referenced by:  ismot  28607