Theorem isismt 25889

Description: Property of being an isometry. Compare with isismty 34229. (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 3414	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		elex 3414	. . . 4 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V)
3		fveq2 6448	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		isismt.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	syl6eqr 2832	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	f1oeq2d 6389	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→(Base‘ℎ)))
7		fveq2 6448	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
8		isismt.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘𝐺)
9	7, 8	syl6eqr 2832	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
10	9	oveqd 6941	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
11	10	eqeq2d 2788	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏)))
12	5, 11	raleqbidv 3326	. . . . . . . 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 3326	. . . . . . 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 624	. . . . . 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 2906	. . . . 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		eqidd 2779	. . . . . . . 8 ⊢ (ℎ = 𝐻 → 𝑓 = 𝑓)
17		eqidd 2779	. . . . . . . 8 ⊢ (ℎ = 𝐻 → 𝐵 = 𝐵)
18		fveq2 6448	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = (Base‘𝐻))
19		isismt.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐻)
20	18, 19	syl6eqr 2832	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = 𝑃)
21	16, 17, 20	f1oeq123d 6388	. . . . . . 7 ⊢ (ℎ = 𝐻 → (𝑓:𝐵–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→𝑃))
22		fveq2 6448	. . . . . . . . . . 11 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = (dist‘𝐻))
23		isismt.m	. . . . . . . . . . 11 ⊢ − = (dist‘𝐻)
24	22, 23	syl6eqr 2832	. . . . . . . . . 10 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = − )
25	24	oveqd 6941	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = ((𝑓‘𝑎) − (𝑓‘𝑏)))
26	25	eqeq1d 2780	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)))
27	26	2ralbidv 3171	. . . . . . 7 ⊢ (ℎ = 𝐻 → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)))
28	21, 27	anbi12d 624	. . . . . 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))))
29	28	abbidv 2906	. . . . 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))})
30		df-ismt 25888	. . . . 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))})
31		ovex 6956	. . . . . 6 ⊢ (𝑃 ↑𝑚 𝐵) ∈ V
32		f1of 6393	. . . . . . . . 9 ⊢ (𝑓:𝐵–1-1-onto→𝑃 → 𝑓:𝐵⟶𝑃)
33	19	fvexi 6462	. . . . . . . . . 10 ⊢ 𝑃 ∈ V
34	4	fvexi 6462	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
35	33, 34	elmap 8171	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑃 ↑𝑚 𝐵) ↔ 𝑓:𝐵⟶𝑃)
36	32, 35	sylibr 226	. . . . . . . 8 ⊢ (𝑓:𝐵–1-1-onto→𝑃 → 𝑓 ∈ (𝑃 ↑𝑚 𝐵))
37	36	adantr 474	. . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)) → 𝑓 ∈ (𝑃 ↑𝑚 𝐵))
38	37	abssi 3898	. . . . . 6 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃 ↑𝑚 𝐵)
39	31, 38	ssexi 5042	. . . . 5 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ∈ V
40	15, 29, 30, 39	ovmpt2 7075	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))})
41	1, 2, 40	syl2an 589	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))})
42	41	eleq2d 2845	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))}))
43		f1of 6393	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→𝑃 → 𝐹:𝐵⟶𝑃)
44		fex 6763	. . . . 5 ⊢ ((𝐹:𝐵⟶𝑃 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
45	43, 34, 44	sylancl 580	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝑃 → 𝐹 ∈ V)
46	45	adantr 474	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)) → 𝐹 ∈ V)
47		f1oeq1 6382	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐵–1-1-onto→𝑃 ↔ 𝐹:𝐵–1-1-onto→𝑃))
48		fveq1 6447	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑎) = (𝐹‘𝑎))
49		fveq1 6447	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑏) = (𝐹‘𝑏))
50	48, 49	oveq12d 6942	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑎) − (𝑓‘𝑏)) = ((𝐹‘𝑎) − (𝐹‘𝑏)))
51	50	eqeq1d 2780	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))
52	51	2ralbidv 3171	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))
53	47, 52	anbi12d 624	. . 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))))
54	46, 53	elab3 3566	. 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)))
55	42, 54	syl6bb 279	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 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {cab 2763  ∀wral 3090  Vcvv 3398  ⟶wf 6133  –1-1-onto→wf1o 6136  ‘cfv 6137  (class class class)co 6924   ↑_𝑚 cmap 8142  Basecbs 16259  distcds 16351  Ismtcismt 25887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-map 8144  df-ismt 25888

This theorem is referenced by:  ismot  25890