Theorem xpsdsval 21996

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 2610	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
7		eqid 2610	. . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
8		eqid 2610	. . . . 5 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))
9	1, 2, 3, 4, 5, 6, 7, 8	xpsval 16055	. . . 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 16056	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
11	6	xpsff1o2 16054	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
12		f1ocnv 6062	. . . . 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		ovex 6577	. . . . 5 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ V
15	14	a1i 11	. . . 4 ⊢ (𝜑 → ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ V)
16		eqid 2610	. . . 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}))))
17		xpsds.p	. . . 4 ⊢ 𝑃 = (dist‘𝑇)
18		xpsds.m	. . . . . 6 ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))
19		xpsds.n	. . . . . 6 ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))
20		xpsds.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
21		xpsds.4	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
22	1, 2, 3, 4, 5, 17, 18, 19, 20, 21	xpsxmetlem 21994	. . . . 5 ⊢ (𝜑 → (dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))))
23		ssid 3587	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ⊆ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
24		xmetres2 21976	. . . . 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}))))
25	22, 23, 24	sylancl 693	. . . 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}))))
26		df-ov 6552	. . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐵) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩)
27		xpsds.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
28		xpsds.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
29	6	xpsfval 16050	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐵) = ◡({𝐴} +𝑐 {𝐵}))
30	27, 28, 29	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐵) = ◡({𝐴} +𝑐 {𝐵}))
31	26, 30	syl5eqr 2658	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}))
32		opelxpi 5072	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
33	27, 28, 32	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
34		f1of 6050	. . . . . . . 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})))
35	11, 34	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
36	35	ffvelrni 6266	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
37	33, 36	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
38	31, 37	eqeltrrd 2689	. . . 4 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
39		df-ov 6552	. . . . . 6 ⊢ (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐷) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩)
40		xpsds.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
41		xpsds.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
42	6	xpsfval 16050	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐷) = ◡({𝐶} +𝑐 {𝐷}))
43	40, 41, 42	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐷) = ◡({𝐶} +𝑐 {𝐷}))
44	39, 43	syl5eqr 2658	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}))
45		opelxpi 5072	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
46	40, 41, 45	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
47	35	ffvelrni 6266	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
48	46, 47	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
49	44, 48	eqeltrrd 2689	. . . 4 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
50	9, 10, 13, 15, 16, 17, 25, 38, 49	imasdsf1o 21989	. . 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})))
51	38, 49	ovresd 6699	. . 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})))
52	50, 51	eqtrd 2644	. 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})))
53		f1ocnvfv 6434	. . . . 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⟩))
54	11, 33, 53	sylancr 694	. . . 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⟩))
55	31, 54	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩)
56		f1ocnvfv 6434	. . . . 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⟩))
57	11, 46, 56	sylancr 694	. . . 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⟩))
58	44, 57	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩)
59	55, 58	oveq12d 6567	. 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⟩))
60		eqid 2610	. . . 4 ⊢ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))
61		fvex 6113	. . . . 5 ⊢ (Scalar‘𝑅) ∈ V
62	61	a1i 11	. . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V)
63		2on 7455	. . . . 5 ⊢ 2𝑜 ∈ On
64	63	a1i 11	. . . 4 ⊢ (𝜑 → 2𝑜 ∈ On)
65		xpscfn 16042	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
66	4, 5, 65	syl2anc 691	. . . 4 ⊢ (𝜑 → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
67	38, 10	eleqtrd 2690	. . . 4 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
68	49, 10	eleqtrd 2690	. . . 4 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
69		eqid 2610	. . . 4 ⊢ (dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))
70	8, 60, 62, 64, 66, 67, 68, 69	prdsdsval 15961	. . 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*, < ))
71		df2o3 7460	. . . . . . . . . . 11 ⊢ 2𝑜 = {∅, 1𝑜}
72	71	rexeqi 3120	. . . . . . . . . 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)))
73		0ex 4718	. . . . . . . . . . 11 ⊢ ∅ ∈ V
74		1on 7454	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ On
75	74	elexi 3186	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ V
76		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ∅ → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = (◡({𝑅} +𝑐 {𝑆})‘∅))
77	76	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → (dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘)) = (dist‘(◡({𝑅} +𝑐 {𝑆})‘∅)))
78		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = (◡({𝐴} +𝑐 {𝐵})‘∅))
79		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑘 = ∅ → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = (◡({𝐶} +𝑐 {𝐷})‘∅))
80	77, 78, 79	oveq123d 6570	. . . . . . . . . . . 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})‘∅)))
81	80	eqeq2d 2620	. . . . . . . . . . 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})‘∅))))
82		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1𝑜 → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = (◡({𝑅} +𝑐 {𝑆})‘1𝑜))
83	82	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1𝑜 → (dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘)) = (dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)))
84		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1𝑜 → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = (◡({𝐴} +𝑐 {𝐵})‘1𝑜))
85		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1𝑜 → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = (◡({𝐶} +𝑐 {𝐷})‘1𝑜))
86	83, 84, 85	oveq123d 6570	. . . . . . . . . . . 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)))
87	86	eqeq2d 2620	. . . . . . . . . . 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))))
88	73, 75, 81, 87	rexpr 4186	. . . . . . . . . 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))))
89	72, 88	bitri 263	. . . . . . . . 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))))
90		xpsc0 16043	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
91	4, 90	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
92	91	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = (dist‘𝑅))
93		xpsc0 16043	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑋 → (◡({𝐴} +𝑐 {𝐵})‘∅) = 𝐴)
94	27, 93	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})‘∅) = 𝐴)
95		xpsc0 16043	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝑋 → (◡({𝐶} +𝑐 {𝐷})‘∅) = 𝐶)
96	40, 95	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡({𝐶} +𝑐 {𝐷})‘∅) = 𝐶)
97	92, 94, 96	oveq123d 6570	. . . . . . . . . . . 12 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)) = (𝐴(dist‘𝑅)𝐶))
98	18	oveqi 6562	. . . . . . . . . . . . 13 ⊢ (𝐴𝑀𝐶) = (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶)
99	27, 40	ovresd 6699	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴((dist‘𝑅) ↾ (𝑋 × 𝑋))𝐶) = (𝐴(dist‘𝑅)𝐶))
100	98, 99	syl5eq 2656	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴𝑀𝐶) = (𝐴(dist‘𝑅)𝐶))
101	97, 100	eqtr4d 2647	. . . . . . . . . . 11 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)) = (𝐴𝑀𝐶))
102	101	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘∅)(dist‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)) ↔ 𝑥 = (𝐴𝑀𝐶)))
103		xpsc1 16044	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑊 → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
104	5, 103	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
105	104	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = (dist‘𝑆))
106		xpsc1 16044	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑌 → (◡({𝐴} +𝑐 {𝐵})‘1𝑜) = 𝐵)
107	28, 106	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})‘1𝑜) = 𝐵)
108		xpsc1 16044	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ 𝑌 → (◡({𝐶} +𝑐 {𝐷})‘1𝑜) = 𝐷)
109	41, 108	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡({𝐶} +𝑐 {𝐷})‘1𝑜) = 𝐷)
110	105, 107, 109	oveq123d 6570	. . . . . . . . . . . 12 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)) = (𝐵(dist‘𝑆)𝐷))
111	19	oveqi 6562	. . . . . . . . . . . . 13 ⊢ (𝐵𝑁𝐷) = (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷)
112	28, 41	ovresd 6699	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵((dist‘𝑆) ↾ (𝑌 × 𝑌))𝐷) = (𝐵(dist‘𝑆)𝐷))
113	111, 112	syl5eq 2656	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵𝑁𝐷) = (𝐵(dist‘𝑆)𝐷))
114	110, 113	eqtr4d 2647	. . . . . . . . . . 11 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)) = (𝐵𝑁𝐷))
115	114	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(dist‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)) ↔ 𝑥 = (𝐵𝑁𝐷)))
116	102, 115	orbi12d 742	. . . . . . . . 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))))
117	89, 116	syl5bb 271	. . . . . . . 8 ⊢ (𝜑 → (∃𝑘 ∈ 2𝑜 𝑥 = ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)) ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷))))
118		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
119		eqid 2610	. . . . . . . . . 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)))
120	119	elrnmpt 5293	. . . . . . . . 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))))
121	118, 120	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)))
122	118	elpr 4146	. . . . . . . 8 ⊢ (𝑥 ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ↔ (𝑥 = (𝐴𝑀𝐶) ∨ 𝑥 = (𝐵𝑁𝐷)))
123	117, 121, 122	3bitr4g 302	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ↔ 𝑥 ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}))
124	123	eqrdv 2608	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) = {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
125	124	uneq1d 3728	. . . . 5 ⊢ (𝜑 → (ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ∪ {0}) = ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}))
126		uncom 3719	. . . . 5 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ∪ {0}) = ({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)})
127	125, 126	syl6eq 2660	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 2𝑜 ↦ ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(dist‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))) ∪ {0}) = ({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}))
128	127	supeq1d 8235	. . 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*, < ))
129		0xr 9965	. . . . . 6 ⊢ 0 ∈ ℝ*
130	129	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ*)
131	130	snssd 4281	. . . 4 ⊢ (𝜑 → {0} ⊆ ℝ*)
132		xmetcl 21946	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) ∈ ℝ*)
133	20, 27, 40, 132	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝐴𝑀𝐶) ∈ ℝ*)
134		xmetcl 21946	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵𝑁𝐷) ∈ ℝ*)
135	21, 28, 41, 134	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝐵𝑁𝐷) ∈ ℝ*)
136		prssi 4293	. . . . 5 ⊢ (((𝐴𝑀𝐶) ∈ ℝ* ∧ (𝐵𝑁𝐷) ∈ ℝ*) → {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ*)
137	133, 135, 136	syl2anc 691	. . . 4 ⊢ (𝜑 → {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ*)
138		xrltso 11850	. . . . . 6 ⊢ < Or ℝ*
139		supsn 8261	. . . . . 6 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
140	138, 129, 139	mp2an 704	. . . . 5 ⊢ sup({0}, ℝ*, < ) = 0
141		supxrcl 12017	. . . . . . 7 ⊢ ({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) ∈ ℝ*)
142	137, 141	syl 17	. . . . . 6 ⊢ (𝜑 → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) ∈ ℝ*)
143		xmetge0 21959	. . . . . . 7 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → 0 ≤ (𝐴𝑀𝐶))
144	20, 27, 40, 143	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴𝑀𝐶))
145		ovex 6577	. . . . . . . 8 ⊢ (𝐴𝑀𝐶) ∈ V
146	145	prid1 4241	. . . . . . 7 ⊢ (𝐴𝑀𝐶) ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}
147		supxrub 12026	. . . . . . 7 ⊢ (({(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* ∧ (𝐴𝑀𝐶) ∈ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}) → (𝐴𝑀𝐶) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
148	137, 146, 147	sylancl 693	. . . . . 6 ⊢ (𝜑 → (𝐴𝑀𝐶) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
149	130, 133, 142, 144, 148	xrletrd 11869	. . . . 5 ⊢ (𝜑 → 0 ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
150	140, 149	syl5eqbr 4618	. . . 4 ⊢ (𝜑 → sup({0}, ℝ*, < ) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
151		supxrun 12018	. . . 4 ⊢ (({0} ⊆ ℝ* ∧ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)} ⊆ ℝ* ∧ sup({0}, ℝ*, < ) ≤ sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < )) → sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ*, < ) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
152	131, 137, 150, 151	syl3anc 1318	. . 3 ⊢ (𝜑 → sup(({0} ∪ {(𝐴𝑀𝐶), (𝐵𝑁𝐷)}), ℝ*, < ) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
153	70, 128, 152	3eqtrd 2648	. 2 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})(dist‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐶} +𝑐 {𝐷})) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
154	52, 59, 153	3eqtr3d 2652	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   Or wor 4958   × cxp 5036  ^◡ccnv 5037  ran crn 5039   ↾ cres 5040  Oncon0 5640   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1_𝑜c1o 7440  2_𝑜c2o 7441  supcsup 8229   +_𝑐 ccda 8872  0cc0 9815  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  Basecbs 15695  Scalarcsca 15771  distcds 15777  X_scprds 15929   ×_s cxps 15989  ∞Metcxmt 19552

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  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-xrs 15985  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-xmet 19560

This theorem is referenced by:  tmsxpsval  22153