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Theorem xmetxp 14827

Description: The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)

**Hypotheses**
Ref	Expression
xmetxp.p	⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))}, ℝ^*, < ))
xmetxp.1	⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
xmetxp.2	⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))

**Assertion**
Ref	Expression
xmetxp	⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))

Distinct variable groups: 𝑢,𝑀,𝑣 𝑢,𝑁,𝑣 𝑢,𝑋,𝑣 𝑢,𝑌,𝑣 Allowed substitution hints: 𝜑(𝑣,𝑢) 𝑃(𝑣,𝑢)

Proof of Theorem xmetxp Dummy variables 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetxp.1	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
2		eqid 2196	. . . . 5 ⊢ (MetOpen‘𝑀) = (MetOpen‘𝑀)
3	2	mopnm 14768	. . . 4 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ (MetOpen‘𝑀))
4	1, 3	syl 14	. . 3 ⊢ (𝜑 → 𝑋 ∈ (MetOpen‘𝑀))
5		xmetxp.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
6		eqid 2196	. . . . 5 ⊢ (MetOpen‘𝑁) = (MetOpen‘𝑁)
7	6	mopnm 14768	. . . 4 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ (MetOpen‘𝑁))
8	5, 7	syl 14	. . 3 ⊢ (𝜑 → 𝑌 ∈ (MetOpen‘𝑁))
9		xpexg 4778	. . 3 ⊢ ((𝑋 ∈ (MetOpen‘𝑀) ∧ 𝑌 ∈ (MetOpen‘𝑁)) → (𝑋 × 𝑌) ∈ V)
10	4, 8, 9	syl2anc 411	. 2 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ V)
11	1	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑀 ∈ (∞Met‘𝑋))
12		xp1st 6232	. . . . . . 7 ⊢ (𝑟 ∈ (𝑋 × 𝑌) → (1^st ‘𝑟) ∈ 𝑋)
13	12	ad2antrl 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (1^st ‘𝑟) ∈ 𝑋)
14		xp1st 6232	. . . . . . 7 ⊢ (𝑠 ∈ (𝑋 × 𝑌) → (1^st ‘𝑠) ∈ 𝑋)
15	14	ad2antll 491	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (1^st ‘𝑠) ∈ 𝑋)
16		xmetcl 14672	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1^st ‘𝑟) ∈ 𝑋 ∧ (1^st ‘𝑠) ∈ 𝑋) → ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ∈ ℝ^*)
17	11, 13, 15, 16	syl3anc 1249	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ∈ ℝ^*)
18	5	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑁 ∈ (∞Met‘𝑌))
19		xp2nd 6233	. . . . . . 7 ⊢ (𝑟 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑟) ∈ 𝑌)
20	19	ad2antrl 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (2^nd ‘𝑟) ∈ 𝑌)
21		xp2nd 6233	. . . . . . 7 ⊢ (𝑠 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑠) ∈ 𝑌)
22	21	ad2antll 491	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (2^nd ‘𝑠) ∈ 𝑌)
23		xmetcl 14672	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2^nd ‘𝑟) ∈ 𝑌 ∧ (2^nd ‘𝑠) ∈ 𝑌) → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^*)
24	18, 20, 22, 23	syl3anc 1249	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^*)
25		xrmaxcl 11434	. . . . 5 ⊢ ((((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ∈ ℝ^* ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^) → sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ∈ ℝ^*)
26	17, 24, 25	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ∈ ℝ^)
27	26	ralrimivva 2579	. . 3 ⊢ (𝜑 → ∀𝑟 ∈ (𝑋 × 𝑌)∀𝑠 ∈ (𝑋 × 𝑌)sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ∈ ℝ^)
28		xmetxp.p	. . . . 5 ⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))}, ℝ^*, < ))
29		fveq2 5561	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (1^st ‘𝑢) = (1^st ‘𝑟))
30	29	oveq1d 5940	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝑟)𝑀(1^st ‘𝑣)))
31		fveq2 5561	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (2^nd ‘𝑢) = (2^nd ‘𝑟))
32	31	oveq1d 5940	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝑟)𝑁(2^nd ‘𝑣)))
33	30, 32	preq12d 3708	. . . . . . 7 ⊢ (𝑢 = 𝑟 → {((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))} = {((1^st ‘𝑟)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑣))})
34	33	supeq1d 7062	. . . . . 6 ⊢ (𝑢 = 𝑟 → sup({((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))}, ℝ^, < ) = sup({((1^st ‘𝑟)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑣))}, ℝ^, < ))
35		fveq2 5561	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → (1^st ‘𝑣) = (1^st ‘𝑠))
36	35	oveq2d 5941	. . . . . . . 8 ⊢ (𝑣 = 𝑠 → ((1^st ‘𝑟)𝑀(1^st ‘𝑣)) = ((1^st ‘𝑟)𝑀(1^st ‘𝑠)))
37		fveq2 5561	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → (2^nd ‘𝑣) = (2^nd ‘𝑠))
38	37	oveq2d 5941	. . . . . . . 8 ⊢ (𝑣 = 𝑠 → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)))
39	36, 38	preq12d 3708	. . . . . . 7 ⊢ (𝑣 = 𝑠 → {((1^st ‘𝑟)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑣))} = {((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))})
40	39	supeq1d 7062	. . . . . 6 ⊢ (𝑣 = 𝑠 → sup({((1^st ‘𝑟)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑣))}, ℝ^, < ) = sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ))
41	34, 40	cbvmpov 6006	. . . . 5 ⊢ (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))}, ℝ^, < )) = (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ))
42	28, 41	eqtri 2217	. . . 4 ⊢ 𝑃 = (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
43	42	fmpo 6268	. . 3 ⊢ (∀𝑟 ∈ (𝑋 × 𝑌)∀𝑠 ∈ (𝑋 × 𝑌)sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ∈ ℝ^ ↔ 𝑃:((𝑋 × 𝑌) × (𝑋 × 𝑌))⟶ℝ^*)
44	27, 43	sylib 122	. 2 ⊢ (𝜑 → 𝑃:((𝑋 × 𝑌) × (𝑋 × 𝑌))⟶ℝ^*)
45		simprl 529	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑟 ∈ (𝑋 × 𝑌))
46		simprr 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑠 ∈ (𝑋 × 𝑌))
47	34, 40, 28	ovmpog 6061	. . . . . . . 8 ⊢ ((𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ∈ ℝ^) → (𝑟𝑃𝑠) = sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
48	45, 46, 26, 47	syl3anc 1249	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) = sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
49	48, 26	eqeltrd 2273	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) ∈ ℝ^*)
50		0xr 8090	. . . . . . 7 ⊢ 0 ∈ ℝ^*
51	50	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ∈ ℝ^*)
52		xrletri3 9896	. . . . . 6 ⊢ (((𝑟𝑃𝑠) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝑟𝑃𝑠) = 0 ↔ ((𝑟𝑃𝑠) ≤ 0 ∧ 0 ≤ (𝑟𝑃𝑠))))
53	49, 51, 52	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) = 0 ↔ ((𝑟𝑃𝑠) ≤ 0 ∧ 0 ≤ (𝑟𝑃𝑠))))
54		xmetge0 14685	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1^st ‘𝑟) ∈ 𝑋 ∧ (1^st ‘𝑠) ∈ 𝑋) → 0 ≤ ((1^st ‘𝑟)𝑀(1^st ‘𝑠)))
55	11, 13, 15, 54	syl3anc 1249	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ≤ ((1^st ‘𝑟)𝑀(1^st ‘𝑠)))
56		xrmax1sup 11435	. . . . . . . . 9 ⊢ ((((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ∈ ℝ^* ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^) → ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ))
57	17, 24, 56	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
58	51, 17, 26, 55, 57	xrletrd 9904	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ≤ sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
59	58, 48	breqtrrd 4062	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ≤ (𝑟𝑃𝑠))
60	59	biantrud 304	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) ≤ 0 ↔ ((𝑟𝑃𝑠) ≤ 0 ∧ 0 ≤ (𝑟𝑃𝑠))))
61	53, 60	bitr4d 191	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) = 0 ↔ (𝑟𝑃𝑠) ≤ 0))
62	48	breq1d 4044	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) ≤ 0 ↔ sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ) ≤ 0))
63		xrmaxlesup 11441	. . . . 5 ⊢ ((((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ∈ ℝ^* ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^* ∧ 0 ∈ ℝ^) → (sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ≤ 0 ↔ (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ 0 ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ 0)))
64	17, 24, 51, 63	syl3anc 1249	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ) ≤ 0 ↔ (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ 0 ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ 0)))
65	61, 62, 64	3bitrd 214	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) = 0 ↔ (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ 0 ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ 0)))
66	55	biantrud 304	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ 0 ↔ (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ 0 ∧ 0 ≤ ((1^st ‘𝑟)𝑀(1^st ‘𝑠)))))
67		xrletri3 9896	. . . . . 6 ⊢ ((((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) = 0 ↔ (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ 0 ∧ 0 ≤ ((1^st ‘𝑟)𝑀(1^st ‘𝑠)))))
68	17, 51, 67	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) = 0 ↔ (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ 0 ∧ 0 ≤ ((1^st ‘𝑟)𝑀(1^st ‘𝑠)))))
69	66, 68	bitr4d 191	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ 0 ↔ ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) = 0))
70		xmetge0 14685	. . . . . . 7 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2^nd ‘𝑟) ∈ 𝑌 ∧ (2^nd ‘𝑠) ∈ 𝑌) → 0 ≤ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)))
71	18, 20, 22, 70	syl3anc 1249	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ≤ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)))
72	71	biantrud 304	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ 0 ↔ (((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ 0 ∧ 0 ≤ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)))))
73		xrletri3 9896	. . . . . 6 ⊢ ((((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) = 0 ↔ (((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ 0 ∧ 0 ≤ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)))))
74	24, 51, 73	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) = 0 ↔ (((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ 0 ∧ 0 ≤ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)))))
75	72, 74	bitr4d 191	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ 0 ↔ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) = 0))
76	69, 75	anbi12d 473	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ 0 ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ 0) ↔ (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) = 0 ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) = 0)))
77		xmeteq0 14679	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1^st ‘𝑟) ∈ 𝑋 ∧ (1^st ‘𝑠) ∈ 𝑋) → (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) = 0 ↔ (1^st ‘𝑟) = (1^st ‘𝑠)))
78	11, 13, 15, 77	syl3anc 1249	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) = 0 ↔ (1^st ‘𝑟) = (1^st ‘𝑠)))
79		xmeteq0 14679	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2^nd ‘𝑟) ∈ 𝑌 ∧ (2^nd ‘𝑠) ∈ 𝑌) → (((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) = 0 ↔ (2^nd ‘𝑟) = (2^nd ‘𝑠)))
80	18, 20, 22, 79	syl3anc 1249	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) = 0 ↔ (2^nd ‘𝑟) = (2^nd ‘𝑠)))
81	78, 80	anbi12d 473	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((((1^st ‘𝑟)𝑀(1^st ‘𝑠)) = 0 ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) = 0) ↔ ((1^st ‘𝑟) = (1^st ‘𝑠) ∧ (2^nd ‘𝑟) = (2^nd ‘𝑠))))
82		xpopth 6243	. . . . 5 ⊢ ((𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌)) → (((1^st ‘𝑟) = (1^st ‘𝑠) ∧ (2^nd ‘𝑟) = (2^nd ‘𝑠)) ↔ 𝑟 = 𝑠))
83	82	adantl 277	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1^st ‘𝑟) = (1^st ‘𝑠) ∧ (2^nd ‘𝑟) = (2^nd ‘𝑠)) ↔ 𝑟 = 𝑠))
84	81, 83	bitrd 188	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((((1^st ‘𝑟)𝑀(1^st ‘𝑠)) = 0 ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) = 0) ↔ 𝑟 = 𝑠))
85	65, 76, 84	3bitrd 214	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) = 0 ↔ 𝑟 = 𝑠))
86	48	3adantr3 1160	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) = sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
87	17	3adantr3 1160	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ∈ ℝ^*)
88	1	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑀 ∈ (∞Met‘𝑋))
89		simpr3 1007	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑡 ∈ (𝑋 × 𝑌))
90		xp1st 6232	. . . . . . . 8 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (1^st ‘𝑡) ∈ 𝑋)
91	89, 90	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (1^st ‘𝑡) ∈ 𝑋)
92		simpr1 1005	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑟 ∈ (𝑋 × 𝑌))
93	92, 12	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (1^st ‘𝑟) ∈ 𝑋)
94		xmetcl 14672	. . . . . . 7 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1^st ‘𝑡) ∈ 𝑋 ∧ (1^st ‘𝑟) ∈ 𝑋) → ((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ∈ ℝ^*)
95	88, 91, 93, 94	syl3anc 1249	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ∈ ℝ^*)
96	15	3adantr3 1160	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (1^st ‘𝑠) ∈ 𝑋)
97		xmetcl 14672	. . . . . . 7 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1^st ‘𝑡) ∈ 𝑋 ∧ (1^st ‘𝑠) ∈ 𝑋) → ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ∈ ℝ^*)
98	88, 91, 96, 97	syl3anc 1249	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ∈ ℝ^*)
99	95, 98	xaddcld 9976	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (((1^st ‘𝑡)𝑀(1^st ‘𝑟)) +_𝑒 ((1^st ‘𝑡)𝑀(1^st ‘𝑠))) ∈ ℝ^*)
100	5	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑁 ∈ (∞Met‘𝑌))
101		xp2nd 6233	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑡) ∈ 𝑌)
102	89, 101	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (2^nd ‘𝑡) ∈ 𝑌)
103	92, 19	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (2^nd ‘𝑟) ∈ 𝑌)
104		xmetcl 14672	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2^nd ‘𝑡) ∈ 𝑌 ∧ (2^nd ‘𝑟) ∈ 𝑌) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ∈ ℝ^*)
105	100, 102, 103, 104	syl3anc 1249	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ∈ ℝ^*)
106		xrmaxcl 11434	. . . . . . . . 9 ⊢ ((((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ∈ ℝ^* ∧ ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ∈ ℝ^) → sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^, < ) ∈ ℝ^*)
107	95, 105, 106	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^, < ) ∈ ℝ^)
108		fveq2 5561	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑡 → (1^st ‘𝑢) = (1^st ‘𝑡))
109		fveq2 5561	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑟 → (1^st ‘𝑣) = (1^st ‘𝑟))
110	108, 109	oveqan12d 5944	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑟) → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝑡)𝑀(1^st ‘𝑟)))
111		fveq2 5561	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑡 → (2^nd ‘𝑢) = (2^nd ‘𝑡))
112		fveq2 5561	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑟 → (2^nd ‘𝑣) = (2^nd ‘𝑟))
113	111, 112	oveqan12d 5944	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑟) → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)))
114	110, 113	preq12d 3708	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑟) → {((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))} = {((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))})
115	114	supeq1d 7062	. . . . . . . . 9 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑟) → sup({((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))}, ℝ^, < ) = sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^, < ))
116	115, 28	ovmpoga 6056	. . . . . . . 8 ⊢ ((𝑡 ∈ (𝑋 × 𝑌) ∧ 𝑟 ∈ (𝑋 × 𝑌) ∧ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^, < ) ∈ ℝ^) → (𝑡𝑃𝑟) = sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^*, < ))
117	89, 92, 107, 116	syl3anc 1249	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑟) = sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^*, < ))
118	117, 107	eqeltrd 2273	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑟) ∈ ℝ^*)
119		simpr2 1006	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑠 ∈ (𝑋 × 𝑌))
120	22	3adantr3 1160	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (2^nd ‘𝑠) ∈ 𝑌)
121		xmetcl 14672	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2^nd ‘𝑡) ∈ 𝑌 ∧ (2^nd ‘𝑠) ∈ 𝑌) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ∈ ℝ^*)
122	100, 102, 120, 121	syl3anc 1249	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ∈ ℝ^*)
123		xrmaxcl 11434	. . . . . . . . 9 ⊢ ((((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ∈ ℝ^* ∧ ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ∈ ℝ^) → sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ∈ ℝ^*)
124	98, 122, 123	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ∈ ℝ^)
125	108, 35	oveqan12d 5944	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑠) → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝑡)𝑀(1^st ‘𝑠)))
126	111, 37	oveqan12d 5944	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑠) → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)))
127	125, 126	preq12d 3708	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑠) → {((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))} = {((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))})
128	127	supeq1d 7062	. . . . . . . . 9 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑠) → sup({((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))}, ℝ^, < ) = sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^, < ))
129	128, 28	ovmpoga 6056	. . . . . . . 8 ⊢ ((𝑡 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ∈ ℝ^) → (𝑡𝑃𝑠) = sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
130	89, 119, 124, 129	syl3anc 1249	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑠) = sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
131	130, 124	eqeltrd 2273	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑠) ∈ ℝ^*)
132	118, 131	xaddcld 9976	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)) ∈ ℝ^*)
133		xmettri2 14681	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ ((1^st ‘𝑡) ∈ 𝑋 ∧ (1^st ‘𝑟) ∈ 𝑋 ∧ (1^st ‘𝑠) ∈ 𝑋)) → ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ (((1^st ‘𝑡)𝑀(1^st ‘𝑟)) +_𝑒 ((1^st ‘𝑡)𝑀(1^st ‘𝑠))))
134	88, 91, 93, 96, 133	syl13anc 1251	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ (((1^st ‘𝑡)𝑀(1^st ‘𝑟)) +_𝑒 ((1^st ‘𝑡)𝑀(1^st ‘𝑠))))
135		xrmax1sup 11435	. . . . . . . 8 ⊢ ((((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ∈ ℝ^* ∧ ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ∈ ℝ^) → ((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ≤ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^, < ))
136	95, 105, 135	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ≤ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^*, < ))
137	136, 117	breqtrrd 4062	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ≤ (𝑡𝑃𝑟))
138		xrmax1sup 11435	. . . . . . . 8 ⊢ ((((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ∈ ℝ^* ∧ ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ∈ ℝ^) → ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ≤ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^, < ))
139	98, 122, 138	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ≤ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
140	139, 130	breqtrrd 4062	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ≤ (𝑡𝑃𝑠))
141		xle2add 9971	. . . . . . 7 ⊢ (((((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ∈ ℝ^* ∧ ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ∈ ℝ^) ∧ ((𝑡𝑃𝑟) ∈ ℝ^ ∧ (𝑡𝑃𝑠) ∈ ℝ^*)) → ((((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ≤ (𝑡𝑃𝑟) ∧ ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ≤ (𝑡𝑃𝑠)) → (((1^st ‘𝑡)𝑀(1^st ‘𝑟)) +_𝑒 ((1^st ‘𝑡)𝑀(1^st ‘𝑠))) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠))))
142	95, 98, 118, 131, 141	syl22anc 1250	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ≤ (𝑡𝑃𝑟) ∧ ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ≤ (𝑡𝑃𝑠)) → (((1^st ‘𝑡)𝑀(1^st ‘𝑟)) +_𝑒 ((1^st ‘𝑡)𝑀(1^st ‘𝑠))) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠))))
143	137, 140, 142	mp2and 433	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (((1^st ‘𝑡)𝑀(1^st ‘𝑟)) +_𝑒 ((1^st ‘𝑡)𝑀(1^st ‘𝑠))) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))
144	87, 99, 132, 134, 143	xrletrd 9904	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))
145	24	3adantr3 1160	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^*)
146	105, 122	xaddcld 9976	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) +_𝑒 ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))) ∈ ℝ^*)
147		xmettri2 14681	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ ((2^nd ‘𝑡) ∈ 𝑌 ∧ (2^nd ‘𝑟) ∈ 𝑌 ∧ (2^nd ‘𝑠) ∈ 𝑌)) → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ (((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) +_𝑒 ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))))
148	100, 102, 103, 120, 147	syl13anc 1251	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ (((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) +_𝑒 ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))))
149		xrmax2sup 11436	. . . . . . . 8 ⊢ ((((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ∈ ℝ^* ∧ ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ∈ ℝ^) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ≤ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^, < ))
150	95, 105, 149	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ≤ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑟)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟))}, ℝ^*, < ))
151	150, 117	breqtrrd 4062	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ≤ (𝑡𝑃𝑟))
152		xrmax2sup 11436	. . . . . . . 8 ⊢ ((((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ∈ ℝ^* ∧ ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ∈ ℝ^) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ≤ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^, < ))
153	98, 122, 152	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ≤ sup({((1^st ‘𝑡)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))}, ℝ^*, < ))
154	153, 130	breqtrrd 4062	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ≤ (𝑡𝑃𝑠))
155		xle2add 9971	. . . . . . 7 ⊢ (((((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ∈ ℝ^* ∧ ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ∈ ℝ^) ∧ ((𝑡𝑃𝑟) ∈ ℝ^ ∧ (𝑡𝑃𝑠) ∈ ℝ^*)) → ((((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ≤ (𝑡𝑃𝑟) ∧ ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ≤ (𝑡𝑃𝑠)) → (((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) +_𝑒 ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠))))
156	105, 122, 118, 131, 155	syl22anc 1250	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ≤ (𝑡𝑃𝑟) ∧ ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ≤ (𝑡𝑃𝑠)) → (((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) +_𝑒 ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠))))
157	151, 154, 156	mp2and 433	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) +_𝑒 ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠))) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))
158	145, 146, 132, 148, 157	xrletrd 9904	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))
159		xrmaxlesup 11441	. . . . 5 ⊢ ((((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ∈ ℝ^* ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^* ∧ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)) ∈ ℝ^) → (sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^, < ) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)) ↔ (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)) ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))))
160	87, 145, 132, 159	syl3anc 1249	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)) ↔ (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)) ∧ ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))))
161	144, 158, 160	mpbir2and 946	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → sup({((1^st ‘𝑟)𝑀(1^st ‘𝑠)), ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠))}, ℝ^*, < ) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))
162	86, 161	eqbrtrd 4056	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))
163	10, 44, 85, 162	isxmetd 14667	1 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 ∀wral 2475 Vcvv 2763 {cpr 3624 class class class wbr 4034 × cxp 4662 ⟶wf 5255 ‘cfv 5259 (class class class)co 5925 ∈ cmpo 5927 1^st c1st 6205 2^nd c2nd 6206 supcsup 7057 0cc0 7896 ℝ^*cxr 8077 < clt 8078 ≤ cle 8079 +_𝑒 cxad 9862 ∞Metcxmet 14168 MetOpencmopn 14173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-map 6718 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-xneg 9864 df-xadd 9865 df-seqfrec 10557 df-exp 10648 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-topgen 12962 df-psmet 14175 df-xmet 14176 df-bl 14178 df-mopn 14179 df-top 14318 df-topon 14331 df-bases 14363

This theorem is referenced by: xmetxpbl 14828 xmettxlem 14829 xmettx 14830 txmetcnp 14838

W3C validator