Theorem isismt 25229

Description: Property of being an isometry. Compare with isismty 32770. (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 3185	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		elex 3185	. . . 4 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V)
3		eqidd 2611	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → 𝑓 = 𝑓)
4		fveq2 6103	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5		isismt.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
6	4, 5	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
7		eqidd 2611	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘ℎ) = (Base‘ℎ))
8	3, 6, 7	f1oeq123d 6046	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→(Base‘ℎ)))
9		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
10		isismt.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘𝐺)
11	9, 10	syl6eqr 2662	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
12	11	oveqd 6566	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑎(dist‘𝑔)𝑏) = (𝑎𝐷𝑏))
13	12	eqeq2d 2620	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏) ↔ ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏)))
14	6, 13	raleqbidv 3129	. . . . . . . 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)))
15	6, 14	raleqbidv 3129	. . . . . . 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)))
16	8, 15	anbi12d 743	. . . . . 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))))
17	16	abbidv 2728	. . . . 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))})
18		eqidd 2611	. . . . . . . 8 ⊢ (ℎ = 𝐻 → 𝑓 = 𝑓)
19		eqidd 2611	. . . . . . . 8 ⊢ (ℎ = 𝐻 → 𝐵 = 𝐵)
20		fveq2 6103	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = (Base‘𝐻))
21		isismt.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐻)
22	20, 21	syl6eqr 2662	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (Base‘ℎ) = 𝑃)
23	18, 19, 22	f1oeq123d 6046	. . . . . . 7 ⊢ (ℎ = 𝐻 → (𝑓:𝐵–1-1-onto→(Base‘ℎ) ↔ 𝑓:𝐵–1-1-onto→𝑃))
24		fveq2 6103	. . . . . . . . . . 11 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = (dist‘𝐻))
25		isismt.m	. . . . . . . . . . 11 ⊢ − = (dist‘𝐻)
26	24, 25	syl6eqr 2662	. . . . . . . . . 10 ⊢ (ℎ = 𝐻 → (dist‘ℎ) = − )
27	26	oveqd 6566	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = ((𝑓‘𝑎) − (𝑓‘𝑏)))
28	27	eqeq1d 2612	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)))
29	28	2ralbidv 2972	. . . . . . 7 ⊢ (ℎ = 𝐻 → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)))
30	23, 29	anbi12d 743	. . . . . 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))))
31	30	abbidv 2728	. . . . 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))})
32		df-ismt 25228	. . . . 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))})
33		ovex 6577	. . . . . 6 ⊢ (𝑃 ↑𝑚 𝐵) ∈ V
34		f1of 6050	. . . . . . . . 9 ⊢ (𝑓:𝐵–1-1-onto→𝑃 → 𝑓:𝐵⟶𝑃)
35		fvex 6113	. . . . . . . . . . 11 ⊢ (Base‘𝐻) ∈ V
36	21, 35	eqeltri 2684	. . . . . . . . . 10 ⊢ 𝑃 ∈ V
37		fvex 6113	. . . . . . . . . . 11 ⊢ (Base‘𝐺) ∈ V
38	5, 37	eqeltri 2684	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
39	36, 38	elmap 7772	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑃 ↑𝑚 𝐵) ↔ 𝑓:𝐵⟶𝑃)
40	34, 39	sylibr 223	. . . . . . . 8 ⊢ (𝑓:𝐵–1-1-onto→𝑃 → 𝑓 ∈ (𝑃 ↑𝑚 𝐵))
41	40	adantr 480	. . . . . . 7 ⊢ ((𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏)) → 𝑓 ∈ (𝑃 ↑𝑚 𝐵))
42	41	abssi 3640	. . . . . 6 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ⊆ (𝑃 ↑𝑚 𝐵)
43	33, 42	ssexi 4731	. . . . 5 ⊢ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))} ∈ V
44	17, 31, 32, 43	ovmpt2 6694	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))})
45	1, 2, 44	syl2an 493	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))})
46	45	eleq2d 2673	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏))}))
47		f1of 6050	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→𝑃 → 𝐹:𝐵⟶𝑃)
48		fex 6394	. . . . 5 ⊢ ((𝐹:𝐵⟶𝑃 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
49	47, 38, 48	sylancl 693	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝑃 → 𝐹 ∈ V)
50	49	adantr 480	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)) → 𝐹 ∈ V)
51		f1oeq1 6040	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐵–1-1-onto→𝑃 ↔ 𝐹:𝐵–1-1-onto→𝑃))
52		fveq1 6102	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑎) = (𝐹‘𝑎))
53		fveq1 6102	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑏) = (𝐹‘𝑏))
54	52, 53	oveq12d 6567	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑎) − (𝑓‘𝑏)) = ((𝐹‘𝑎) − (𝐹‘𝑏)))
55	54	eqeq1d 2612	. . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))
56	55	2ralbidv 2972	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑓‘𝑎) − (𝑓‘𝑏)) = (𝑎𝐷𝑏) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))
57	51, 56	anbi12d 743	. . 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))))
58	50, 57	elab3 3327	. 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)))
59	46, 58	syl6bb 275	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  Basecbs 15695  distcds 15777  Ismtcismt 25227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ismt 25228

This theorem is referenced by:  ismot  25230