Theorem xpsdsval 22186

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 2622	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
7		eqid 2622	. . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
8		eqid 2622	. . . . 5 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))
9	1, 2, 3, 4, 5, 6, 7, 8	xpsval 16232	. . . 4 ⊢ (𝜑 → 𝑇 = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
10	1, 2, 3, 4, 5, 6, 7, 8	xpslem 16233	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
11	6	xpsff1o2 16231	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
12		f1ocnv 6149	. . . . 5 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌))
13	11, 12	mp1i 13	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌))
14		ovexd 6680	. . . 4 ⊢ (𝜑 → ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ V)
15		eqid 2622	. . . 4 ⊢ ((dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))) = ((dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {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 22184	. . . . 5 ⊢ (𝜑 → (dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))))
22		ssid 3624	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ⊆ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
23		xmetres2 22166	. . . . 5 ⊢ (((dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))) ∧ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ⊆ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))) → ((dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))))
24	21, 22, 23	sylancl 694	. . . 4 ⊢ (𝜑 → ((dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))))
25		df-ov 6653	. . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐵) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩)
26		xpsds.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
27		xpsds.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
28	6	xpsfval 16227	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐵) = ◡({𝐴} +𝑐 {𝐵}))
29	26, 27, 28	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐵) = ◡({𝐴} +𝑐 {𝐵}))
30	25, 29	syl5eqr 2670	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}))
31		opelxpi 5148	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
32	26, 27, 31	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
33		f1of 6137	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
34	11, 33	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
35	34	ffvelrni 6358	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
36	32, 35	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
37	30, 36	eqeltrrd 2702	. . . 4 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
38		df-ov 6653	. . . . . 6 ⊢ (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐷) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩)
39		xpsds.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
40		xpsds.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
41	6	xpsfval 16227	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐷) = ◡({𝐶} +𝑐 {𝐷}))
42	39, 40, 41	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐷) = ◡({𝐶} +𝑐 {𝐷}))
43	38, 42	syl5eqr 2670	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}))
44		opelxpi 5148	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
45	39, 40, 44	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
46	34	ffvelrni 6358	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
47	45, 46	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
48	43, 47	eqeltrrd 2702	. . . 4 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
49	9, 10, 13, 14, 15, 16, 24, 37, 48	imasdsf1o 22179	. . 3 ⊢ (𝜑 → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵}))𝑃(◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷}))) = (◡({𝐴} +𝑐 {𝐵})((dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))))◡({𝐶} +𝑐 {𝐷})))
50	37, 48	ovresd 6801	. . 3 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})((dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ↾ (ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) × ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))))◡({𝐶} +𝑐 {𝐷})) = (◡({𝐴} +𝑐 {𝐵})(dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐶} +𝑐 {𝐷})))
51	49, 50	eqtrd 2656	. 2 ⊢ (𝜑 → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵}))𝑃(◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷}))) = (◡({𝐴} +𝑐 {𝐵})(dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐶} +𝑐 {𝐷})))
52		f1ocnvfv 6534	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩))
53	11, 32, 52	sylancr 695	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩))
54	30, 53	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩)
55		f1ocnvfv 6534	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩))
56	11, 45, 55	sylancr 695	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩))
57	43, 56	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩)
58	54, 57	oveq12d 6668	. 2 ⊢ (𝜑 → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵}))𝑃(◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷}))) = (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩))
59		eqid 2622	. . . 4 ⊢ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))
60		fvexd 6203	. . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V)
61		2on 7568	. . . . 5 ⊢ 2𝑜 ∈ On
62	61	a1i 11	. . . 4 ⊢ (𝜑 → 2𝑜 ∈ On)
63		xpscfn 16219	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
64	4, 5, 63	syl2anc 693	. . . 4 ⊢ (𝜑 → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
65	37, 10	eleqtrd 2703	. . . 4 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
66	48, 10	eleqtrd 2703	. . . 4 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
67		eqid 2622	. . . 4 ⊢ (dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))
68	8, 59, 60, 62, 64, 65, 66, 67	prdsdsval 16138	. . 3 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})(dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐶} +𝑐 {𝐷})) = sup((ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ∪ {0}), ℝ*, < ))
69		df2o3 7573	. . . . . . . . . . 11 ⊢ 2𝑜 = {∅, 1𝑜}
70	69	rexeqi 3143	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 2𝑜 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) ↔ ∃𝑘 ∈ {∅, 1𝑜}𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)))
71		0ex 4790	. . . . . . . . . . 11 ⊢ ∅ ∈ V
72		1on 7567	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ On
73	72	elexi 3213	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ V
74		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ∅ → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = (◡({𝑅} +𝑐 {𝑆})‘∅))
75	74	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → (dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘)) = (dist‘(◡({𝑅} +𝑐 {𝑆})‘∅)))
76		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = (◡({𝐴} +𝑐 {𝐵})‘∅))
77		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = (◡({𝐶} +𝑐 {𝐷})‘∅))
78	75, 76, 77	oveq123d 6671	. . . . . . . . . . . 12 ⊢ (𝑘 = ∅ → ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) = ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)))
79	78	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) ↔ 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅))))
80		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1𝑜 → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = (◡({𝑅} +𝑐 {𝑆})‘1𝑜))
81	80	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1𝑜 → (dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘)) = (dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)))
82		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1𝑜 → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = (◡({𝐴} +𝑐 {𝐵})‘1𝑜))
83		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1𝑜 → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = (◡({𝐶} +𝑐 {𝐷})‘1𝑜))
84	81, 82, 83	oveq123d 6671	. . . . . . . . . . . 12 ⊢ (𝑘 = 1𝑜 → ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) = ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)))
85	84	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑘 = 1𝑜 → (𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) ↔ 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜))))
86	71, 73, 79, 85	rexpr 4239	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ {∅, 1𝑜}𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) ↔ (𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)) ∨ 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜))))
87	70, 86	bitri 264	. . . . . . . . 9 ⊢ (∃𝑘 ∈ 2𝑜 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) ↔ (𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)) ∨ 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜))))
88		xpsc0 16220	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
89	4, 88	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
90	89	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = (dist‘𝑅))
91		xpsc0 16220	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑋 → (◡({𝐴} +𝑐 {𝐵})‘∅) = 𝐴)
92	26, 91	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})‘∅) = 𝐴)
93		xpsc0 16220	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝑋 → (◡({𝐶} +𝑐 {𝐷})‘∅) = 𝐶)
94	39, 93	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡({𝐶} +𝑐 {𝐷})‘∅) = 𝐶)
95	90, 92, 94	oveq123d 6671	. . . . . . . . . . . 12 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)) = (𝐴(dist‘𝑅)𝐶))
96	17	oveqi 6663	. . . . . . . . . . . . 13 ⊢ (𝐴𝑀𝐶) = (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶)
97	26, 39	ovresd 6801	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶) = (𝐴(dist‘𝑅)𝐶))
98	96, 97	syl5eq 2668	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝑀𝐶) = (𝐴(dist‘𝑅)𝐶))
99	95, 98	eqtr4d 2659	. . . . . . . . . . 11 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)) = (𝐴𝑀𝐶))
100	99	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)) ↔ 𝑥 = (𝐴𝑀𝐶)))
101		xpsc1 16221	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑊 → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
102	5, 101	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
103	102	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = (dist‘𝑆))
104		xpsc1 16221	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑌 → (◡({𝐴} +𝑐 {𝐵})‘1𝑜) = 𝐵)
105	27, 104	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})‘1𝑜) = 𝐵)
106		xpsc1 16221	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ 𝑌 → (◡({𝐶} +𝑐 {𝐷})‘1𝑜) = 𝐷)
107	40, 106	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡({𝐶} +𝑐 {𝐷})‘1𝑜) = 𝐷)
108	103, 105, 107	oveq123d 6671	. . . . . . . . . . . 12 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)) = (𝐵(dist‘𝑆)𝐷))
109	18	oveqi 6663	. . . . . . . . . . . . 13 ⊢ (𝐵𝑁𝐷) = (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷)
110	27, 40	ovresd 6801	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷) = (𝐵(dist‘𝑆)𝐷))
111	109, 110	syl5eq 2668	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵𝑁𝐷) = (𝐵(dist‘𝑆)𝐷))
112	108, 111	eqtr4d 2659	. . . . . . . . . . 11 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)) = (𝐵𝑁𝐷))
113	112	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)) ↔ 𝑥 = (𝐵𝑁𝐷)))
114	100, 113	orbi12d 746	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)) ∨ 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜))) ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷))))
115	87, 114	syl5bb 272	. . . . . . . 8 ⊢ (𝜑 → (∃𝑘 ∈ 2𝑜 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷))))
116		vex 3203	. . . . . . . . 9 ⊢ 𝑥 ∈ V
117		eqid 2622	. . . . . . . . . 10 ⊢ (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) = (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)))
118	117	elrnmpt 5372	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ↔ ∃𝑘 ∈ 2𝑜 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))))
119	116, 118	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ↔ ∃𝑘 ∈ 2𝑜 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)))
120	116	elpr 4198	. . . . . . . 8 ⊢ (𝑥 ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷)))
121	115, 119, 120	3bitr4g 303	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ↔ 𝑥 ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}))
122	121	eqrdv 2620	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) = {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
123	122	uneq1d 3766	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ∪ {0}) = ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}))
124		uncom 3757	. . . . 5 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}) = ({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
125	123, 124	syl6eq 2672	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ∪ {0}) = ({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}))
126	125	supeq1d 8352	. . 3 ⊢ (𝜑 → sup((ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ∪ {0}), ℝ*, < ) = sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ*, < ))
127		0xr 10086	. . . . . 6 ⊢ 0 ∈ ℝ*
128	127	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ*)
129	128	snssd 4340	. . . 4 ⊢ (𝜑 → {0} ⊆ ℝ*)
130		xmetcl 22136	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) ∈ ℝ*)
131	19, 26, 39, 130	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (𝐴𝑀𝐶) ∈ ℝ*)
132		xmetcl 22136	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵𝑁𝐷) ∈ ℝ*)
133	20, 27, 40, 132	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (𝐵𝑁𝐷) ∈ ℝ*)
134		prssi 4353	. . . . 5 ⊢ (((𝐴𝑀𝐶) ∈ ℝ* ∧ (𝐵𝑁𝐷) ∈ ℝ*) → {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ*)
135	131, 133, 134	syl2anc 693	. . . 4 ⊢ (𝜑 → {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ*)
136		xrltso 11974	. . . . . 6 ⊢ < Or ℝ*
137		supsn 8378	. . . . . 6 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
138	136, 127, 137	mp2an 708	. . . . 5 ⊢ sup({0}, ℝ*, < ) = 0
139		supxrcl 12145	. . . . . . 7 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) ∈ ℝ*)
140	135, 139	syl 17	. . . . . 6 ⊢ (𝜑 → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) ∈ ℝ*)
141		xmetge0 22149	. . . . . . 7 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → 0 ≤ (𝐴𝑀𝐶))
142	19, 26, 39, 141	syl3anc 1326	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴𝑀𝐶))
143		ovex 6678	. . . . . . . 8 ⊢ (𝐴𝑀𝐶) ∈ V
144	143	prid1 4297	. . . . . . 7 ⊢ (𝐴𝑀𝐶) ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}
145		supxrub 12154	. . . . . . 7 ⊢ (({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* ∧ (𝐴𝑀𝐶) ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}) → (𝐴𝑀𝐶) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
146	135, 144, 145	sylancl 694	. . . . . 6 ⊢ (𝜑 → (𝐴𝑀𝐶) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
147	128, 131, 140, 142, 146	xrletrd 11993	. . . . 5 ⊢ (𝜑 → 0 ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
148	138, 147	syl5eqbr 4688	. . . 4 ⊢ (𝜑 → sup({0}, ℝ*, < ) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
149		supxrun 12146	. . . 4 ⊢ (({0} ⊆ ℝ* ∧ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* ∧ sup({0}, ℝ*, < ) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < )) → sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ*, < ) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
150	129, 135, 148, 149	syl3anc 1326	. . 3 ⊢ (𝜑 → sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ*, < ) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
151	68, 126, 150	3eqtrd 2660	. 2 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})(dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐶} +𝑐 {𝐷})) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
152	51, 58, 151	3eqtr3d 2664	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729   Or wor 5034   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1_𝑜c1o 7553  2_𝑜c2o 7554  supcsup 8346   +_𝑐 ccda 8989  0cc0 9936  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  Basecbs 15857  Scalarcsca 15944  distcds 15950  X_scprds 16106   ×_s cxps 16166  ∞Metcxmt 19731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-xrs 16162  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-xmet 19739

This theorem is referenced by:  tmsxpsval  22343