Theorem xpsdsval 23640

Description: Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
xpsds.t	⊢ 𝑇 = (𝑅 ×s 𝑆)
xpsds.x	⊢ 𝑋 = (Base‘𝑅)
xpsds.y	⊢ 𝑌 = (Base‘𝑆)
xpsds.1	⊢ (𝜑 → 𝑅 ∈ 𝑉)
xpsds.2	⊢ (𝜑 → 𝑆 ∈ 𝑊)
xpsds.p	⊢ 𝑃 = (dist‘𝑇)
xpsds.m	⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))
xpsds.n	⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))
xpsds.3	⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
xpsds.4	⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
xpsds.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
xpsds.b	⊢ (𝜑 → 𝐵 ∈ 𝑌)
xpsds.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
xpsds.d	⊢ (𝜑 → 𝐷 ∈ 𝑌)

Assertion

Ref	Expression
xpsdsval	⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))

Proof of Theorem xpsdsval

Dummy variables 𝑥 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsds.t	. . . . 5 ⊢ 𝑇 = (𝑅 ×s 𝑆)
2		xpsds.x	. . . . 5 ⊢ 𝑋 = (Base‘𝑅)
3		xpsds.y	. . . . 5 ⊢ 𝑌 = (Base‘𝑆)
4		xpsds.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
5		xpsds.2	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
6		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
7		eqid 2736	. . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
8		eqid 2736	. . . . 5 ⊢ ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) = ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
9	1, 2, 3, 4, 5, 6, 7, 8	xpsval 17378	. . . 4 ⊢ (𝜑 → 𝑇 = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
10	1, 2, 3, 4, 5, 6, 7, 8	xpsrnbas 17379	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
11	6	xpsff1o2 17377	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
12		f1ocnv 6779	. . . . 5 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})–1-1-onto→(𝑋 × 𝑌))
13	11, 12	mp1i 13	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})–1-1-onto→(𝑋 × 𝑌))
14		ovexd 7372	. . . 4 ⊢ (𝜑 → ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) ∈ V)
15		eqid 2736	. . . 4 ⊢ ((dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))) = ((dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})))
16		xpsds.p	. . . 4 ⊢ 𝑃 = (dist‘𝑇)
17		xpsds.m	. . . . . 6 ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))
18		xpsds.n	. . . . . 6 ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))
19		xpsds.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
20		xpsds.4	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
21	1, 2, 3, 4, 5, 16, 17, 18, 19, 20	xpsxmetlem 23638	. . . . 5 ⊢ (𝜑 → (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})))
22		ssid 3954	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) ⊆ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
23		xmetres2 23620	. . . . 5 ⊢ (((dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})) ∧ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) ⊆ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})) → ((dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})))
24	21, 22, 23	sylancl 586	. . . 4 ⊢ (𝜑 → ((dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})))
25		df-ov 7340	. . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐵) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐴, 𝐵⟩)
26		xpsds.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
27		xpsds.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
28	6	xpsfval 17374	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
29	26, 27, 28	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
30	25, 29	eqtr3id 2790	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
31	26, 27	opelxpd 5658	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
32		f1of 6767	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
33	11, 32	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
34	33	ffvelcdmi 7016	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
35	31, 34	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
36	30, 35	eqeltrrd 2838	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
37		df-ov 7340	. . . . . 6 ⊢ (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐷) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐶, 𝐷⟩)
38		xpsds.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
39		xpsds.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
40	6	xpsfval 17374	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
41	38, 39, 40	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
42	37, 41	eqtr3id 2790	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
43	38, 39	opelxpd 5658	. . . . . 6 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
44	33	ffvelcdmi 7016	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
45	43, 44	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
46	42, 45	eqeltrrd 2838	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
47	9, 10, 13, 14, 15, 16, 24, 36, 46	imasdsf1o 23633	. . 3 ⊢ (𝜑 → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})𝑃(◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ((dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}))
48	36, 46	ovresd 7501	. . 3 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ((dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}))
49	47, 48	eqtrd 2776	. 2 ⊢ (𝜑 → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})𝑃(◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}))
50		f1ocnvfv 7206	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
51	11, 31, 50	sylancr 587	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
52	30, 51	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩)
53		f1ocnvfv 7206	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
54	11, 43, 53	sylancr 587	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
55	42, 54	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩)
56	52, 55	oveq12d 7355	. 2 ⊢ (𝜑 → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})𝑃(◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩))
57		eqid 2736	. . . 4 ⊢ (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
58		fvexd 6840	. . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V)
59		2on 8381	. . . . 5 ⊢ 2o ∈ On
60	59	a1i 11	. . . 4 ⊢ (𝜑 → 2o ∈ On)
61		fnpr2o 17365	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
62	4, 5, 61	syl2anc 584	. . . 4 ⊢ (𝜑 → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
63	36, 10	eleqtrd 2839	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
64	46, 10	eleqtrd 2839	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
65		eqid 2736	. . . 4 ⊢ (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
66	8, 57, 58, 60, 62, 63, 64, 65	prdsdsval 17286	. . 3 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = sup((ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) ∪ {0}), ℝ*, < ))
67		df2o3 8375	. . . . . . . . . . 11 ⊢ 2o = {∅, 1o}
68	67	rexeqi 3308	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 2o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) ↔ ∃𝑘 ∈ {∅, 1o}𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)))
69		0ex 5251	. . . . . . . . . . 11 ⊢ ∅ ∈ V
70		1oex 8377	. . . . . . . . . . 11 ⊢ 1o ∈ V
71		2fveq3 6830	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘)) = (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅)))
72		fveq2 6825	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅))
73		fveq2 6825	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅))
74	71, 72, 73	oveq123d 7358	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)))
75	74	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) ↔ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅))))
76		2fveq3 6830	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1o → (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘)) = (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o)))
77		fveq2 6825	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1o → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o))
78		fveq2 6825	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1o → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o))
79	76, 77, 78	oveq123d 7358	. . . . . . . . . . . 12 ⊢ (𝑘 = 1o → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o)))
80	79	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑘 = 1o → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) ↔ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o))))
81	69, 70, 75, 80	rexpr 4649	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ {∅, 1o}𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) ↔ (𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)) ∨ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o))))
82	68, 81	bitri 274	. . . . . . . . 9 ⊢ (∃𝑘 ∈ 2o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) ↔ (𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)) ∨ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o))))
83		fvpr0o 17367	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅) = 𝑅)
84	4, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅) = 𝑅)
85	84	fveq2d 6829	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅)) = (dist‘𝑅))
86		fvpr0o 17367	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑋 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅) = 𝐴)
87	26, 86	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅) = 𝐴)
88		fvpr0o 17367	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝑋 → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅) = 𝐶)
89	38, 88	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅) = 𝐶)
90	85, 87, 89	oveq123d 7358	. . . . . . . . . . . 12 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)) = (𝐴(dist‘𝑅)𝐶))
91	17	oveqi 7350	. . . . . . . . . . . . 13 ⊢ (𝐴𝑀𝐶) = (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶)
92	26, 38	ovresd 7501	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶) = (𝐴(dist‘𝑅)𝐶))
93	91, 92	eqtrid 2788	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝑀𝐶) = (𝐴(dist‘𝑅)𝐶))
94	90, 93	eqtr4d 2779	. . . . . . . . . . 11 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)) = (𝐴𝑀𝐶))
95	94	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)) ↔ 𝑥 = (𝐴𝑀𝐶)))
96		fvpr1o 17368	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑊 → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o) = 𝑆)
97	5, 96	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o) = 𝑆)
98	97	fveq2d 6829	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o)) = (dist‘𝑆))
99		fvpr1o 17368	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑌 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o) = 𝐵)
100	27, 99	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o) = 𝐵)
101		fvpr1o 17368	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ 𝑌 → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o) = 𝐷)
102	39, 101	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o) = 𝐷)
103	98, 100, 102	oveq123d 7358	. . . . . . . . . . . 12 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o)) = (𝐵(dist‘𝑆)𝐷))
104	18	oveqi 7350	. . . . . . . . . . . . 13 ⊢ (𝐵𝑁𝐷) = (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷)
105	27, 39	ovresd 7501	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷) = (𝐵(dist‘𝑆)𝐷))
106	104, 105	eqtrid 2788	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵𝑁𝐷) = (𝐵(dist‘𝑆)𝐷))
107	103, 106	eqtr4d 2779	. . . . . . . . . . 11 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o)) = (𝐵𝑁𝐷))
108	107	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o)) ↔ 𝑥 = (𝐵𝑁𝐷)))
109	95, 108	orbi12d 916	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)) ∨ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o))) ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷))))
110	82, 109	bitrid 282	. . . . . . . 8 ⊢ (𝜑 → (∃𝑘 ∈ 2o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷))))
111		eqid 2736	. . . . . . . . . 10 ⊢ (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) = (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)))
112	111	elrnmpt 5897	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) ↔ ∃𝑘 ∈ 2o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))))
113	112	elv 3447	. . . . . . . 8 ⊢ (𝑥 ∈ ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) ↔ ∃𝑘 ∈ 2o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)))
114		vex 3445	. . . . . . . . 9 ⊢ 𝑥 ∈ V
115	114	elpr 4596	. . . . . . . 8 ⊢ (𝑥 ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷)))
116	110, 113, 115	3bitr4g 313	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) ↔ 𝑥 ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}))
117	116	eqrdv 2734	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) = {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
118	117	uneq1d 4109	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) ∪ {0}) = ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}))
119		uncom 4100	. . . . 5 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}) = ({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
120	118, 119	eqtrdi 2792	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) ∪ {0}) = ({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}))
121	120	supeq1d 9303	. . 3 ⊢ (𝜑 → sup((ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) ∪ {0}), ℝ*, < ) = sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ*, < ))
122		0xr 11123	. . . . . 6 ⊢ 0 ∈ ℝ*
123	122	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ*)
124	123	snssd 4756	. . . 4 ⊢ (𝜑 → {0} ⊆ ℝ*)
125		xmetcl 23590	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) ∈ ℝ*)
126	19, 26, 38, 125	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝐴𝑀𝐶) ∈ ℝ*)
127		xmetcl 23590	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵𝑁𝐷) ∈ ℝ*)
128	20, 27, 39, 127	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝐵𝑁𝐷) ∈ ℝ*)
129	126, 128	prssd 4769	. . . 4 ⊢ (𝜑 → {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ*)
130		xrltso 12976	. . . . . 6 ⊢ < Or ℝ*
131		supsn 9329	. . . . . 6 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
132	130, 122, 131	mp2an 689	. . . . 5 ⊢ sup({0}, ℝ*, < ) = 0
133		supxrcl 13150	. . . . . . 7 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) ∈ ℝ*)
134	129, 133	syl 17	. . . . . 6 ⊢ (𝜑 → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) ∈ ℝ*)
135		xmetge0 23603	. . . . . . 7 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → 0 ≤ (𝐴𝑀𝐶))
136	19, 26, 38, 135	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴𝑀𝐶))
137		ovex 7370	. . . . . . . 8 ⊢ (𝐴𝑀𝐶) ∈ V
138	137	prid1 4710	. . . . . . 7 ⊢ (𝐴𝑀𝐶) ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}
139		supxrub 13159	. . . . . . 7 ⊢ (({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* ∧ (𝐴𝑀𝐶) ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}) → (𝐴𝑀𝐶) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
140	129, 138, 139	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝐴𝑀𝐶) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
141	123, 126, 134, 136, 140	xrletrd 12997	. . . . 5 ⊢ (𝜑 → 0 ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
142	132, 141	eqbrtrid 5127	. . . 4 ⊢ (𝜑 → sup({0}, ℝ*, < ) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
143		supxrun 13151	. . . 4 ⊢ (({0} ⊆ ℝ* ∧ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* ∧ sup({0}, ℝ*, < ) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < )) → sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ*, < ) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
144	124, 129, 142, 143	syl3anc 1370	. . 3 ⊢ (𝜑 → sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ*, < ) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
145	66, 121, 144	3eqtrd 2780	. 2 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
146	49, 56, 145	3eqtr3d 2784	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 844   = wceq 1540   ∈ wcel 2105  ∃wrex 3070  Vcvv 3441   ∪ cun 3896   ⊆ wss 3898  ∅c0 4269  {csn 4573  {cpr 4575  ⟨cop 4579   class class class wbr 5092   ↦ cmpt 5175   Or wor 5531   × cxp 5618  ^◡ccnv 5619  ran crn 5621   ↾ cres 5622  Oncon0 6302   Fn wfn 6474  ⟶wf 6475  –1-1-onto→wf1o 6478  ‘cfv 6479  (class class class)co 7337   ∈ cmpo 7339  1_oc1o 8360  2_oc2o 8361  supcsup 9297  0cc0 10972  ℝ^*cxr 11109   < clt 11110   ≤ cle 11111  Basecbs 17009  Scalarcsca 17062  distcds 17068  X_scprds 17253   ×_s cxps 17314  ∞Metcxmet 20688

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-of 7595  df-om 7781  df-1st 7899  df-2nd 7900  df-supp 8048  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-2o 8368  df-er 8569  df-map 8688  df-ixp 8757  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-fsupp 9227  df-sup 9299  df-inf 9300  df-oi 9367  df-card 9796  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-7 12142  df-8 12143  df-9 12144  df-n0 12335  df-z 12421  df-dec 12539  df-uz 12684  df-rp 12832  df-xneg 12949  df-xadd 12950  df-xmul 12951  df-icc 13187  df-fz 13341  df-fzo 13484  df-seq 13823  df-hash 14146  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-mulr 17073  df-sca 17075  df-vsca 17076  df-ip 17077  df-tset 17078  df-ple 17079  df-ds 17081  df-hom 17083  df-cco 17084  df-0g 17249  df-gsum 17250  df-prds 17255  df-xrs 17310  df-imas 17316  df-xps 17318  df-mre 17392  df-mrc 17393  df-acs 17395  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-submnd 18528  df-mulg 18797  df-cntz 19019  df-cmn 19483  df-xmet 20696

This theorem is referenced by:  tmsxpsval  23800