Theorem xpsdsval 23442

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 2738	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
7		eqid 2738	. . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
8		eqid 2738	. . . . 5 ⊢ ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) = ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
9	1, 2, 3, 4, 5, 6, 7, 8	xpsval 17198	. . . 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 17199	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
11	6	xpsff1o2 17197	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
12		f1ocnv 6712	. . . . 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 7290	. . . 4 ⊢ (𝜑 → ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) ∈ V)
15		eqid 2738	. . . 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 23440	. . . . 5 ⊢ (𝜑 → (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})))
22		ssid 3939	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) ⊆ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
23		xmetres2 23422	. . . . 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 585	. . . 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 7258	. . . . . 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 17194	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
29	26, 27, 28	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
30	25, 29	eqtr3id 2793	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
31	26, 27	opelxpd 5618	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
32		f1of 6700	. . . . . . . 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	ffvelrni 6942	. . . . . 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 2840	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
37		df-ov 7258	. . . . . 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 17194	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
41	38, 39, 40	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
42	37, 41	eqtr3id 2793	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
43	38, 39	opelxpd 5618	. . . . . 6 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
44	33	ffvelrni 6942	. . . . . 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 2840	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
47	9, 10, 13, 14, 15, 16, 24, 36, 46	imasdsf1o 23435	. . 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 7417	. . 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 2778	. 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 7131	. . . . 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 586	. . . 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 7131	. . . . 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 586	. . . 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 7273	. 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 2738	. . . 4 ⊢ (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
58		fvexd 6771	. . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V)
59		2on 8275	. . . . 5 ⊢ 2o ∈ On
60	59	a1i 11	. . . 4 ⊢ (𝜑 → 2o ∈ On)
61		fnpr2o 17185	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
62	4, 5, 61	syl2anc 583	. . . 4 ⊢ (𝜑 → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
63	36, 10	eleqtrd 2841	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
64	46, 10	eleqtrd 2841	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
65		eqid 2738	. . . 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 17106	. . 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 8282	. . . . . . . . . . 11 ⊢ 2o = {∅, 1o}
68	67	rexeqi 3338	. . . . . . . . . 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 5226	. . . . . . . . . . 11 ⊢ ∅ ∈ V
70		1oex 8280	. . . . . . . . . . 11 ⊢ 1o ∈ V
71		2fveq3 6761	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘)) = (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅)))
72		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅))
73		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅))
74	71, 72, 73	oveq123d 7276	. . . . . . . . . . . 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 2749	. . . . . . . . . . 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 6761	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1o → (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘)) = (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o)))
77		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1o → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o))
78		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1o → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o))
79	76, 77, 78	oveq123d 7276	. . . . . . . . . . . 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 2749	. . . . . . . . . . 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 4634	. . . . . . . . . 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 17187	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅) = 𝑅)
84	4, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅) = 𝑅)
85	84	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅)) = (dist‘𝑅))
86		fvpr0o 17187	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑋 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅) = 𝐴)
87	26, 86	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅) = 𝐴)
88		fvpr0o 17187	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝑋 → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅) = 𝐶)
89	38, 88	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅) = 𝐶)
90	85, 87, 89	oveq123d 7276	. . . . . . . . . . . 12 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)) = (𝐴(dist‘𝑅)𝐶))
91	17	oveqi 7268	. . . . . . . . . . . . 13 ⊢ (𝐴𝑀𝐶) = (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶)
92	26, 38	ovresd 7417	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶) = (𝐴(dist‘𝑅)𝐶))
93	91, 92	eqtrid 2790	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝑀𝐶) = (𝐴(dist‘𝑅)𝐶))
94	90, 93	eqtr4d 2781	. . . . . . . . . . 11 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)) = (𝐴𝑀𝐶))
95	94	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘∅)) ↔ 𝑥 = (𝐴𝑀𝐶)))
96		fvpr1o 17188	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑊 → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o) = 𝑆)
97	5, 96	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o) = 𝑆)
98	97	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o)) = (dist‘𝑆))
99		fvpr1o 17188	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑌 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o) = 𝐵)
100	27, 99	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o) = 𝐵)
101		fvpr1o 17188	. . . . . . . . . . . . . 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 7276	. . . . . . . . . . . 12 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o)) = (𝐵(dist‘𝑆)𝐷))
104	18	oveqi 7268	. . . . . . . . . . . . 13 ⊢ (𝐵𝑁𝐷) = (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷)
105	27, 39	ovresd 7417	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷) = (𝐵(dist‘𝑆)𝐷))
106	104, 105	eqtrid 2790	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵𝑁𝐷) = (𝐵(dist‘𝑆)𝐷))
107	103, 106	eqtr4d 2781	. . . . . . . . . . 11 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o)) = (𝐵𝑁𝐷))
108	107	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘1o)) ↔ 𝑥 = (𝐵𝑁𝐷)))
109	95, 108	orbi12d 915	. . . . . . . . 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	syl5bb 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 2738	. . . . . . . . . 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 5854	. . . . . . . . 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 3428	. . . . . . . 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 3426	. . . . . . . . 9 ⊢ 𝑥 ∈ V
115	114	elpr 4581	. . . . . . . 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 2736	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) = {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
118	117	uneq1d 4092	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) ∪ {0}) = ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}))
119		uncom 4083	. . . . 5 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}) = ({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
120	118, 119	eqtrdi 2795	. . . 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 9135	. . 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 10953	. . . . . 6 ⊢ 0 ∈ ℝ*
123	122	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ*)
124	123	snssd 4739	. . . 4 ⊢ (𝜑 → {0} ⊆ ℝ*)
125		xmetcl 23392	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) ∈ ℝ*)
126	19, 26, 38, 125	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (𝐴𝑀𝐶) ∈ ℝ*)
127		xmetcl 23392	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵𝑁𝐷) ∈ ℝ*)
128	20, 27, 39, 127	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (𝐵𝑁𝐷) ∈ ℝ*)
129	126, 128	prssd 4752	. . . 4 ⊢ (𝜑 → {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ*)
130		xrltso 12804	. . . . . 6 ⊢ < Or ℝ*
131		supsn 9161	. . . . . 6 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
132	130, 122, 131	mp2an 688	. . . . 5 ⊢ sup({0}, ℝ*, < ) = 0
133		supxrcl 12978	. . . . . . 7 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) ∈ ℝ*)
134	129, 133	syl 17	. . . . . 6 ⊢ (𝜑 → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) ∈ ℝ*)
135		xmetge0 23405	. . . . . . 7 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → 0 ≤ (𝐴𝑀𝐶))
136	19, 26, 38, 135	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴𝑀𝐶))
137		ovex 7288	. . . . . . . 8 ⊢ (𝐴𝑀𝐶) ∈ V
138	137	prid1 4695	. . . . . . 7 ⊢ (𝐴𝑀𝐶) ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}
139		supxrub 12987	. . . . . . 7 ⊢ (({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* ∧ (𝐴𝑀𝐶) ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}) → (𝐴𝑀𝐶) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
140	129, 138, 139	sylancl 585	. . . . . 6 ⊢ (𝜑 → (𝐴𝑀𝐶) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
141	123, 126, 134, 136, 140	xrletrd 12825	. . . . 5 ⊢ (𝜑 → 0 ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
142	132, 141	eqbrtrid 5105	. . . 4 ⊢ (𝜑 → sup({0}, ℝ*, < ) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
143		supxrun 12979	. . . 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 1369	. . 3 ⊢ (𝜑 → sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ*, < ) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
145	66, 121, 144	3eqtrd 2782	. 2 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
146	49, 56, 145	3eqtr3d 2786	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  {cpr 4560  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153   Or wor 5493   × cxp 5578  ^◡ccnv 5579  ran crn 5581   ↾ cres 5582  Oncon0 6251   Fn wfn 6413  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1_oc1o 8260  2_oc2o 8261  supcsup 9129  0cc0 10802  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941  Basecbs 16840  Scalarcsca 16891  distcds 16897  X_scprds 17073   ×_s cxps 17134  ∞Metcxmet 20495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-icc 13015  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-hom 16912  df-cco 16913  df-0g 17069  df-gsum 17070  df-prds 17075  df-xrs 17130  df-imas 17136  df-xps 17138  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-xmet 20503

This theorem is referenced by:  tmsxpsval  23600