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Theorem xmetxp 15364

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 2232	. . . . 5 ⊢ (MetOpen‘𝑀) = (MetOpen‘𝑀)
3	2	mopnm 15305	. . . 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 2232	. . . . 5 ⊢ (MetOpen‘𝑁) = (MetOpen‘𝑁)
7	6	mopnm 15305	. . . 4 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ (MetOpen‘𝑁))
8	5, 7	syl 14	. . 3 ⊢ (𝜑 → 𝑌 ∈ (MetOpen‘𝑁))
9		xpexg 4863	. . 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 6358	. . . . . . 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 6358	. . . . . . 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 15209	. . . . . 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 1274	. . . . 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 6359	. . . . . . 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 6359	. . . . . . 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 15209	. . . . . 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 1274	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^*)
25		xrmaxcl 11933	. . . . 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 2624	. . 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 5669	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (1^st ‘𝑢) = (1^st ‘𝑟))
30	29	oveq1d 6064	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝑟)𝑀(1^st ‘𝑣)))
31		fveq2 5669	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (2^nd ‘𝑢) = (2^nd ‘𝑟))
32	31	oveq1d 6064	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝑟)𝑁(2^nd ‘𝑣)))
33	30, 32	preq12d 3775	. . . . . . 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 7277	. . . . . 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 5669	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → (1^st ‘𝑣) = (1^st ‘𝑠))
36	35	oveq2d 6065	. . . . . . . 8 ⊢ (𝑣 = 𝑠 → ((1^st ‘𝑟)𝑀(1^st ‘𝑣)) = ((1^st ‘𝑟)𝑀(1^st ‘𝑠)))
37		fveq2 5669	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → (2^nd ‘𝑣) = (2^nd ‘𝑠))
38	37	oveq2d 6065	. . . . . . . 8 ⊢ (𝑣 = 𝑠 → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)))
39	36, 38	preq12d 3775	. . . . . . 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 7277	. . . . . 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 6132	. . . . 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 2253	. . . 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 6396	. . 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 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑟 ∈ (𝑋 × 𝑌))
46		simprr 533	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑠 ∈ (𝑋 × 𝑌))
47	34, 40, 28	ovmpog 6187	. . . . . . . 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 1274	. . . . . . 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 2309	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) ∈ ℝ^*)
50		0xr 8319	. . . . . . 7 ⊢ 0 ∈ ℝ^*
51	50	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ∈ ℝ^*)
52		xrletri3 10136	. . . . . 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 15222	. . . . . . . . 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 1274	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ≤ ((1^st ‘𝑟)𝑀(1^st ‘𝑠)))
56		xrmax1sup 11934	. . . . . . . . 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 10144	. . . . . . 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 4136	. . . . . 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 4118	. . . 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 11940	. . . . 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 1274	. . . 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 10136	. . . . . 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 15222	. . . . . . 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 1274	. . . . . 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 10136	. . . . . 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 15216	. . . . . 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 1274	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1^st ‘𝑟)𝑀(1^st ‘𝑠)) = 0 ↔ (1^st ‘𝑟) = (1^st ‘𝑠)))
79		xmeteq0 15216	. . . . . 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 1274	. . . . 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 6369	. . . . 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 1185	. . 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 1185	. . . . 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 1032	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑡 ∈ (𝑋 × 𝑌))
90		xp1st 6358	. . . . . . . 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 1030	. . . . . . . 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 15209	. . . . . . 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 1274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ∈ ℝ^*)
96	15	3adantr3 1185	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (1^st ‘𝑠) ∈ 𝑋)
97		xmetcl 15209	. . . . . . 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 1274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ∈ ℝ^*)
99	95, 98	xaddcld 10216	. . . . 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 6359	. . . . . . . . . . 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 15209	. . . . . . . . . 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 1274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ∈ ℝ^*)
106		xrmaxcl 11933	. . . . . . . . 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 5669	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑡 → (1^st ‘𝑢) = (1^st ‘𝑡))
109		fveq2 5669	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑟 → (1^st ‘𝑣) = (1^st ‘𝑟))
110	108, 109	oveqan12d 6068	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑟) → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝑡)𝑀(1^st ‘𝑟)))
111		fveq2 5669	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑡 → (2^nd ‘𝑢) = (2^nd ‘𝑡))
112		fveq2 5669	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑟 → (2^nd ‘𝑣) = (2^nd ‘𝑟))
113	111, 112	oveqan12d 6068	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑟) → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)))
114	110, 113	preq12d 3775	. . . . . . . . . 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 7277	. . . . . . . . 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 6182	. . . . . . . 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 1274	. . . . . . 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 2309	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑟) ∈ ℝ^*)
119		simpr2 1031	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑠 ∈ (𝑋 × 𝑌))
120	22	3adantr3 1185	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (2^nd ‘𝑠) ∈ 𝑌)
121		xmetcl 15209	. . . . . . . . . 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 1274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ∈ ℝ^*)
123		xrmaxcl 11933	. . . . . . . . 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 6068	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑠) → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝑡)𝑀(1^st ‘𝑠)))
126	111, 37	oveqan12d 6068	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑠) → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)))
127	125, 126	preq12d 3775	. . . . . . . . . 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 7277	. . . . . . . . 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 6182	. . . . . . . 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 1274	. . . . . . 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 2309	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑠) ∈ ℝ^*)
132	118, 131	xaddcld 10216	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)) ∈ ℝ^*)
133		xmettri2 15218	. . . . . 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 1276	. . . . 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 11934	. . . . . . . 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 4136	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑟)) ≤ (𝑡𝑃𝑟))
138		xrmax1sup 11934	. . . . . . . 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 4136	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑡)𝑀(1^st ‘𝑠)) ≤ (𝑡𝑃𝑠))
141		xle2add 10211	. . . . . . 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 1275	. . . . . 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 10144	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1^st ‘𝑟)𝑀(1^st ‘𝑠)) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))
145	24	3adantr3 1185	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ∈ ℝ^*)
146	105, 122	xaddcld 10216	. . . . 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 15218	. . . . . 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 1276	. . . . 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 11935	. . . . . . . 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 4136	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑟)) ≤ (𝑡𝑃𝑟))
152		xrmax2sup 11935	. . . . . . . 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 4136	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑡)𝑁(2^nd ‘𝑠)) ≤ (𝑡𝑃𝑠))
155		xle2add 10211	. . . . . . 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 1275	. . . . . 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 10144	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2^nd ‘𝑟)𝑁(2^nd ‘𝑠)) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))
159		xrmaxlesup 11940	. . . . 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 1274	. . . 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 953	. . 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 4130	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) ≤ ((𝑡𝑃𝑟) +_𝑒 (𝑡𝑃𝑠)))
163	10, 44, 85, 162	isxmetd 15204	1 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 ∀wral 2520 Vcvv 2812 {cpr 3689 class class class wbr 4108 × cxp 4746 ⟶wf 5347 ‘cfv 5351 (class class class)co 6049 ∈ cmpo 6051 1^st c1st 6331 2^nd c2nd 6332 supcsup 7272 0cc0 8126 ℝ^*cxr 8306 < clt 8307 ≤ cle 8308 +_𝑒 cxad 10102 ∞Metcxmet 14676 MetOpencmopn 14681

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-mulrcl 8225 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-0lt1 8232 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-precex 8236 ax-cnre 8237 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240 ax-pre-apti 8241 ax-pre-ltadd 8242 ax-pre-mulgt0 8243 ax-pre-mulext 8244 ax-arch 8245 ax-caucvg 8246

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-isom 5360 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-frec 6621 df-map 6883 df-sup 7274 df-inf 7275 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313 df-sub 8445 df-neg 8446 df-reap 8848 df-ap 8855 df-div 8946 df-inn 9237 df-2 9295 df-3 9296 df-4 9297 df-n0 9496 df-z 9577 df-uz 9853 df-q 9951 df-rp 9986 df-xneg 10104 df-xadd 10105 df-seqfrec 10809 df-exp 10900 df-cj 11523 df-re 11524 df-im 11525 df-rsqrt 11679 df-abs 11680 df-topgen 13465 df-psmet 14683 df-xmet 14684 df-bl 14686 df-mopn 14687 df-top 14855 df-topon 14868 df-bases 14900

This theorem is referenced by: xmetxpbl 15365 xmettxlem 15366 xmettx 15367 txmetcnp 15375

W3C validator