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Theorem xpsdsval 23894

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 2732	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})
7		eqid 2732	. . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
8		eqid 2732	. . . . 5 ⊢ ((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}) = ((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})
9	1, 2, 3, 4, 5, 6, 7, 8	xpsval 17518	. . . 4 ⊢ (𝜑 → 𝑇 = (^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) “_s ((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})))
10	1, 2, 3, 4, 5, 6, 7, 8	xpsrnbas 17519	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) = (Base‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})))
11	6	xpsff1o2 17517	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})
12		f1ocnv 6845	. . . . 5 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) → ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})–1-1-onto→(𝑋 × 𝑌))
13	11, 12	mp1i 13	. . . 4 ⊢ (𝜑 → ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})–1-1-onto→(𝑋 × 𝑌))
14		ovexd 7446	. . . 4 ⊢ (𝜑 → ((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}) ∈ V)
15		eqid 2732	. . . 4 ⊢ ((dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))) = ((dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 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 23892	. . . . 5 ⊢ (𝜑 → (dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})))
22		ssid 4004	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) ⊆ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})
23		xmetres2 23874	. . . . 5 ⊢ (((dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})) ∧ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) ⊆ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})) → ((dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})))
24	21, 22, 23	sylancl 586	. . . 4 ⊢ (𝜑 → ((dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})))
25		df-ov 7414	. . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})𝐵) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐴, 𝐵⟩)
26		xpsds.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
27		xpsds.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
28	6	xpsfval 17514	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})𝐵) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩})
29	26, 27, 28	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})𝐵) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩})
30	25, 29	eqtr3id 2786	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩})
31	26, 27	opelxpd 5715	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
32		f1of 6833	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))
33	11, 32	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})
34	33	ffvelcdmi 7085	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))
35	31, 34	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))
36	30, 35	eqeltrrd 2834	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))
37		df-ov 7414	. . . . . 6 ⊢ (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})𝐷) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐶, 𝐷⟩)
38		xpsds.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
39		xpsds.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
40	6	xpsfval 17514	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})𝐷) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})
41	38, 39, 40	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})𝐷) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})
42	37, 41	eqtr3id 2786	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})
43	38, 39	opelxpd 5715	. . . . . 6 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
44	33	ffvelcdmi 7085	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))
45	43, 44	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))
46	42, 45	eqeltrrd 2834	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))
47	9, 10, 13, 14, 15, 16, 24, 36, 46	imasdsf1o 23887	. . 3 ⊢ (𝜑 → ((^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩})𝑃(^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})) = ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ((dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}))
48	36, 46	ovresd 7576	. . 3 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ((dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}))){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}))
49	47, 48	eqtrd 2772	. 2 ⊢ (𝜑 → ((^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩})𝑃(^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})) = ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}))
50		f1ocnvfv 7278	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} → (^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
51	11, 31, 50	sylancr 587	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} → (^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
52	30, 51	mpd 15	. . 3 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩)
53		f1ocnvfv 7278	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} → (^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
54	11, 43, 53	sylancr 587	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} → (^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
55	42, 54	mpd 15	. . 3 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩)
56	52, 55	oveq12d 7429	. 2 ⊢ (𝜑 → ((^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩})𝑃(^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})‘{⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩})) = (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩))
57		eqid 2732	. . . 4 ⊢ (Base‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) = (Base‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}))
58		fvexd 6906	. . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V)
59		2on 8482	. . . . 5 ⊢ 2_o ∈ On
60	59	a1i 11	. . . 4 ⊢ (𝜑 → 2_o ∈ On)
61		fnpr2o 17505	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩} Fn 2_o)
62	4, 5, 61	syl2anc 584	. . . 4 ⊢ (𝜑 → {⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩} Fn 2_o)
63	36, 10	eleqtrd 2835	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} ∈ (Base‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})))
64	46, 10	eleqtrd 2835	. . . 4 ⊢ (𝜑 → {⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩} ∈ (Base‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})))
65		eqid 2732	. . . 4 ⊢ (dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})) = (dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}))
66	8, 57, 58, 60, 62, 63, 64, 65	prdsdsval 17426	. . 3 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = sup((ran (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) ∪ {0}), ℝ^*, < ))
67		df2o3 8476	. . . . . . . . . . 11 ⊢ 2_o = {∅, 1_o}
68	67	rexeqi 3324	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 2_o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)) ↔ ∃𝑘 ∈ {∅, 1_o}𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)))
69		0ex 5307	. . . . . . . . . . 11 ⊢ ∅ ∈ V
70		1oex 8478	. . . . . . . . . . 11 ⊢ 1_o ∈ V
71		2fveq3 6896	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → (dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘)) = (dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅)))
72		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘) = ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅))
73		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘) = ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅))
74	71, 72, 73	oveq123d 7432	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)) = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅)))
75	74	eqeq2d 2743	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)) ↔ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅))))
76		2fveq3 6896	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1_o → (dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘)) = (dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o)))
77		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1_o → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘) = ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o))
78		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1_o → ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘) = ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o))
79	76, 77, 78	oveq123d 7432	. . . . . . . . . . . 12 ⊢ (𝑘 = 1_o → (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)) = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o)))
80	79	eqeq2d 2743	. . . . . . . . . . 11 ⊢ (𝑘 = 1_o → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)) ↔ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o))))
81	69, 70, 75, 80	rexpr 4705	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ {∅, 1_o}𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)) ↔ (𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅)) ∨ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o))))
82	68, 81	bitri 274	. . . . . . . . 9 ⊢ (∃𝑘 ∈ 2_o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)) ↔ (𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅)) ∨ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o))))
83		fvpr0o 17507	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → ({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅) = 𝑅)
84	4, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅) = 𝑅)
85	84	fveq2d 6895	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅)) = (dist‘𝑅))
86		fvpr0o 17507	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑋 → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅) = 𝐴)
87	26, 86	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅) = 𝐴)
88		fvpr0o 17507	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝑋 → ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅) = 𝐶)
89	38, 88	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅) = 𝐶)
90	85, 87, 89	oveq123d 7432	. . . . . . . . . . . 12 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅)) = (𝐴(dist‘𝑅)𝐶))
91	17	oveqi 7424	. . . . . . . . . . . . 13 ⊢ (𝐴𝑀𝐶) = (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶)
92	26, 38	ovresd 7576	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶) = (𝐴(dist‘𝑅)𝐶))
93	91, 92	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝑀𝐶) = (𝐴(dist‘𝑅)𝐶))
94	90, 93	eqtr4d 2775	. . . . . . . . . . 11 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅)) = (𝐴𝑀𝐶))
95	94	eqeq2d 2743	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅)) ↔ 𝑥 = (𝐴𝑀𝐶)))
96		fvpr1o 17508	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑊 → ({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o) = 𝑆)
97	5, 96	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o) = 𝑆)
98	97	fveq2d 6895	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o)) = (dist‘𝑆))
99		fvpr1o 17508	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑌 → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o) = 𝐵)
100	27, 99	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o) = 𝐵)
101		fvpr1o 17508	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ 𝑌 → ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o) = 𝐷)
102	39, 101	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o) = 𝐷)
103	98, 100, 102	oveq123d 7432	. . . . . . . . . . . 12 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o)) = (𝐵(dist‘𝑆)𝐷))
104	18	oveqi 7424	. . . . . . . . . . . . 13 ⊢ (𝐵𝑁𝐷) = (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷)
105	27, 39	ovresd 7576	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷) = (𝐵(dist‘𝑆)𝐷))
106	104, 105	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵𝑁𝐷) = (𝐵(dist‘𝑆)𝐷))
107	103, 106	eqtr4d 2775	. . . . . . . . . . 11 ⊢ (𝜑 → (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o)) = (𝐵𝑁𝐷))
108	107	eqeq2d 2743	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o)) ↔ 𝑥 = (𝐵𝑁𝐷)))
109	95, 108	orbi12d 917	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘∅)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘∅))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘∅)) ∨ 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘1_o)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘1_o))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘1_o))) ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷))))
110	82, 109	bitrid 282	. . . . . . . 8 ⊢ (𝜑 → (∃𝑘 ∈ 2_o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)) ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷))))
111		eqid 2732	. . . . . . . . . 10 ⊢ (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) = (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)))
112	111	elrnmpt 5955	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) ↔ ∃𝑘 ∈ 2_o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))))
113	112	elv 3480	. . . . . . . 8 ⊢ (𝑥 ∈ ran (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) ↔ ∃𝑘 ∈ 2_o 𝑥 = (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘)))
114		vex 3478	. . . . . . . . 9 ⊢ 𝑥 ∈ V
115	114	elpr 4651	. . . . . . . 8 ⊢ (𝑥 ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷)))
116	110, 113, 115	3bitr4g 313	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ran (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) ↔ 𝑥 ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}))
117	116	eqrdv 2730	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) = {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
118	117	uneq1d 4162	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) ∪ {0}) = ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}))
119		uncom 4153	. . . . 5 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}) = ({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
120	118, 119	eqtrdi 2788	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) ∪ {0}) = ({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}))
121	120	supeq1d 9443	. . 3 ⊢ (𝜑 → sup((ran (𝑘 ∈ 2_o ↦ (({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩}‘𝑘)(dist‘({⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}‘𝑘))) ∪ {0}), ℝ^, < ) = sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ^, < ))
122		0xr 11263	. . . . . 6 ⊢ 0 ∈ ℝ^*
123	122	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ^*)
124	123	snssd 4812	. . . 4 ⊢ (𝜑 → {0} ⊆ ℝ^*)
125		xmetcl 23844	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) ∈ ℝ^*)
126	19, 26, 38, 125	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (𝐴𝑀𝐶) ∈ ℝ^*)
127		xmetcl 23844	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵𝑁𝐷) ∈ ℝ^*)
128	20, 27, 39, 127	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (𝐵𝑁𝐷) ∈ ℝ^*)
129	126, 128	prssd 4825	. . . 4 ⊢ (𝜑 → {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ^*)
130		xrltso 13122	. . . . . 6 ⊢ < Or ℝ^*
131		supsn 9469	. . . . . 6 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
132	130, 122, 131	mp2an 690	. . . . 5 ⊢ sup({0}, ℝ^*, < ) = 0
133		supxrcl 13296	. . . . . . 7 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ^* → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ^, < ) ∈ ℝ^)
134	129, 133	syl 17	. . . . . 6 ⊢ (𝜑 → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ^, < ) ∈ ℝ^)
135		xmetge0 23857	. . . . . . 7 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → 0 ≤ (𝐴𝑀𝐶))
136	19, 26, 38, 135	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴𝑀𝐶))
137		ovex 7444	. . . . . . . 8 ⊢ (𝐴𝑀𝐶) ∈ V
138	137	prid1 4766	. . . . . . 7 ⊢ (𝐴𝑀𝐶) ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}
139		supxrub 13305	. . . . . . 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 13143	. . . . 5 ⊢ (𝜑 → 0 ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ^*, < ))
142	132, 141	eqbrtrid 5183	. . . 4 ⊢ (𝜑 → sup({0}, ℝ^, < ) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ^, < ))
143		supxrun 13297	. . . 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 1371	. . 3 ⊢ (𝜑 → sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ^, < ) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ^, < ))
145	66, 121, 144	3eqtrd 2776	. 2 ⊢ (𝜑 → ({⟨∅, 𝐴⟩, ⟨1_o, 𝐵⟩} (dist‘((Scalar‘𝑅)X_s{⟨∅, 𝑅⟩, ⟨1_o, 𝑆⟩})){⟨∅, 𝐶⟩, ⟨1_o, 𝐷⟩}) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ^*, < ))
146	49, 56, 145	3eqtr3d 2780	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 Vcvv 3474 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 {csn 4628 {cpr 4630 ⟨cop 4634 class class class wbr 5148 ↦ cmpt 5231 Or wor 5587 × cxp 5674 ^◡ccnv 5675 ran crn 5677 ↾ cres 5678 Oncon0 6364 Fn wfn 6538 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 1_oc1o 8461 2_oc2o 8462 supcsup 9437 0cc0 11112 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 Basecbs 17146 Scalarcsca 17202 distcds 17208 X_scprds 17393 ×_s cxps 17454 ∞Metcxmet 20935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-icc 13333 df-fz 13487 df-fzo 13630 df-seq 13969 df-hash 14293 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-hom 17223 df-cco 17224 df-0g 17389 df-gsum 17390 df-prds 17395 df-xrs 17450 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-xmet 20943

This theorem is referenced by: tmsxpsval 24054

W3C validator