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Theorem xmetxp 12571
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({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))
xmetxp.1 (𝜑𝑀 ∈ (∞Met‘𝑋))
xmetxp.2 (𝜑𝑁 ∈ (∞Met‘𝑌))
Assertion
Ref Expression
xmetxp (𝜑𝑃 ∈ (∞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.
StepHypRef Expression
1 xmetxp.1 . . . 4 (𝜑𝑀 ∈ (∞Met‘𝑋))
2 eqid 2115 . . . . 5 (MetOpen‘𝑀) = (MetOpen‘𝑀)
32mopnm 12512 . . . 4 (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ (MetOpen‘𝑀))
41, 3syl 14 . . 3 (𝜑𝑋 ∈ (MetOpen‘𝑀))
5 xmetxp.2 . . . 4 (𝜑𝑁 ∈ (∞Met‘𝑌))
6 eqid 2115 . . . . 5 (MetOpen‘𝑁) = (MetOpen‘𝑁)
76mopnm 12512 . . . 4 (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ (MetOpen‘𝑁))
85, 7syl 14 . . 3 (𝜑𝑌 ∈ (MetOpen‘𝑁))
9 xpexg 4621 . . 3 ((𝑋 ∈ (MetOpen‘𝑀) ∧ 𝑌 ∈ (MetOpen‘𝑁)) → (𝑋 × 𝑌) ∈ V)
104, 8, 9syl2anc 406 . 2 (𝜑 → (𝑋 × 𝑌) ∈ V)
111adantr 272 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑀 ∈ (∞Met‘𝑋))
12 xp1st 6029 . . . . . . 7 (𝑟 ∈ (𝑋 × 𝑌) → (1st𝑟) ∈ 𝑋)
1312ad2antrl 479 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (1st𝑟) ∈ 𝑋)
14 xp1st 6029 . . . . . . 7 (𝑠 ∈ (𝑋 × 𝑌) → (1st𝑠) ∈ 𝑋)
1514ad2antll 480 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (1st𝑠) ∈ 𝑋)
16 xmetcl 12416 . . . . . 6 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝑟) ∈ 𝑋 ∧ (1st𝑠) ∈ 𝑋) → ((1st𝑟)𝑀(1st𝑠)) ∈ ℝ*)
1711, 13, 15, 16syl3anc 1199 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((1st𝑟)𝑀(1st𝑠)) ∈ ℝ*)
185adantr 272 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑁 ∈ (∞Met‘𝑌))
19 xp2nd 6030 . . . . . . 7 (𝑟 ∈ (𝑋 × 𝑌) → (2nd𝑟) ∈ 𝑌)
2019ad2antrl 479 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (2nd𝑟) ∈ 𝑌)
21 xp2nd 6030 . . . . . . 7 (𝑠 ∈ (𝑋 × 𝑌) → (2nd𝑠) ∈ 𝑌)
2221ad2antll 480 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (2nd𝑠) ∈ 𝑌)
23 xmetcl 12416 . . . . . 6 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝑟) ∈ 𝑌 ∧ (2nd𝑠) ∈ 𝑌) → ((2nd𝑟)𝑁(2nd𝑠)) ∈ ℝ*)
2418, 20, 22, 23syl3anc 1199 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((2nd𝑟)𝑁(2nd𝑠)) ∈ ℝ*)
25 xrmaxcl 10961 . . . . 5 ((((1st𝑟)𝑀(1st𝑠)) ∈ ℝ* ∧ ((2nd𝑟)𝑁(2nd𝑠)) ∈ ℝ*) → sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ∈ ℝ*)
2617, 24, 25syl2anc 406 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ∈ ℝ*)
2726ralrimivva 2489 . . 3 (𝜑 → ∀𝑟 ∈ (𝑋 × 𝑌)∀𝑠 ∈ (𝑋 × 𝑌)sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ∈ ℝ*)
28 xmetxp.p . . . . 5 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))
29 fveq2 5387 . . . . . . . . 9 (𝑢 = 𝑟 → (1st𝑢) = (1st𝑟))
3029oveq1d 5755 . . . . . . . 8 (𝑢 = 𝑟 → ((1st𝑢)𝑀(1st𝑣)) = ((1st𝑟)𝑀(1st𝑣)))
31 fveq2 5387 . . . . . . . . 9 (𝑢 = 𝑟 → (2nd𝑢) = (2nd𝑟))
3231oveq1d 5755 . . . . . . . 8 (𝑢 = 𝑟 → ((2nd𝑢)𝑁(2nd𝑣)) = ((2nd𝑟)𝑁(2nd𝑣)))
3330, 32preq12d 3576 . . . . . . 7 (𝑢 = 𝑟 → {((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))} = {((1st𝑟)𝑀(1st𝑣)), ((2nd𝑟)𝑁(2nd𝑣))})
3433supeq1d 6840 . . . . . 6 (𝑢 = 𝑟 → sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ) = sup({((1st𝑟)𝑀(1st𝑣)), ((2nd𝑟)𝑁(2nd𝑣))}, ℝ*, < ))
35 fveq2 5387 . . . . . . . . 9 (𝑣 = 𝑠 → (1st𝑣) = (1st𝑠))
3635oveq2d 5756 . . . . . . . 8 (𝑣 = 𝑠 → ((1st𝑟)𝑀(1st𝑣)) = ((1st𝑟)𝑀(1st𝑠)))
37 fveq2 5387 . . . . . . . . 9 (𝑣 = 𝑠 → (2nd𝑣) = (2nd𝑠))
3837oveq2d 5756 . . . . . . . 8 (𝑣 = 𝑠 → ((2nd𝑟)𝑁(2nd𝑣)) = ((2nd𝑟)𝑁(2nd𝑠)))
3936, 38preq12d 3576 . . . . . . 7 (𝑣 = 𝑠 → {((1st𝑟)𝑀(1st𝑣)), ((2nd𝑟)𝑁(2nd𝑣))} = {((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))})
4039supeq1d 6840 . . . . . 6 (𝑣 = 𝑠 → sup({((1st𝑟)𝑀(1st𝑣)), ((2nd𝑟)𝑁(2nd𝑣))}, ℝ*, < ) = sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ))
4134, 40cbvmpov 5817 . . . . 5 (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < )) = (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ))
4228, 41eqtri 2136 . . . 4 𝑃 = (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ))
4342fmpo 6065 . . 3 (∀𝑟 ∈ (𝑋 × 𝑌)∀𝑠 ∈ (𝑋 × 𝑌)sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ∈ ℝ*𝑃:((𝑋 × 𝑌) × (𝑋 × 𝑌))⟶ℝ*)
4427, 43sylib 121 . 2 (𝜑𝑃:((𝑋 × 𝑌) × (𝑋 × 𝑌))⟶ℝ*)
45 simprl 503 . . . . . . . 8 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑟 ∈ (𝑋 × 𝑌))
46 simprr 504 . . . . . . . 8 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 𝑠 ∈ (𝑋 × 𝑌))
4734, 40, 28ovmpog 5871 . . . . . . . 8 ((𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ∈ ℝ*) → (𝑟𝑃𝑠) = sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ))
4845, 46, 26, 47syl3anc 1199 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) = sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ))
4948, 26eqeltrd 2192 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) ∈ ℝ*)
50 0xr 7776 . . . . . . 7 0 ∈ ℝ*
5150a1i 9 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ∈ ℝ*)
52 xrletri3 9528 . . . . . 6 (((𝑟𝑃𝑠) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑟𝑃𝑠) = 0 ↔ ((𝑟𝑃𝑠) ≤ 0 ∧ 0 ≤ (𝑟𝑃𝑠))))
5349, 51, 52syl2anc 406 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) = 0 ↔ ((𝑟𝑃𝑠) ≤ 0 ∧ 0 ≤ (𝑟𝑃𝑠))))
54 xmetge0 12429 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝑟) ∈ 𝑋 ∧ (1st𝑠) ∈ 𝑋) → 0 ≤ ((1st𝑟)𝑀(1st𝑠)))
5511, 13, 15, 54syl3anc 1199 . . . . . . . 8 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ≤ ((1st𝑟)𝑀(1st𝑠)))
56 xrmax1sup 10962 . . . . . . . . 9 ((((1st𝑟)𝑀(1st𝑠)) ∈ ℝ* ∧ ((2nd𝑟)𝑁(2nd𝑠)) ∈ ℝ*) → ((1st𝑟)𝑀(1st𝑠)) ≤ sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ))
5717, 24, 56syl2anc 406 . . . . . . . 8 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((1st𝑟)𝑀(1st𝑠)) ≤ sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ))
5851, 17, 26, 55, 57xrletrd 9535 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ≤ sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ))
5958, 48breqtrrd 3924 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ≤ (𝑟𝑃𝑠))
6059biantrud 300 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) ≤ 0 ↔ ((𝑟𝑃𝑠) ≤ 0 ∧ 0 ≤ (𝑟𝑃𝑠))))
6153, 60bitr4d 190 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) = 0 ↔ (𝑟𝑃𝑠) ≤ 0))
6248breq1d 3907 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) ≤ 0 ↔ sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ≤ 0))
63 xrmaxlesup 10968 . . . . 5 ((((1st𝑟)𝑀(1st𝑠)) ∈ ℝ* ∧ ((2nd𝑟)𝑁(2nd𝑠)) ∈ ℝ* ∧ 0 ∈ ℝ*) → (sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ≤ 0 ↔ (((1st𝑟)𝑀(1st𝑠)) ≤ 0 ∧ ((2nd𝑟)𝑁(2nd𝑠)) ≤ 0)))
6417, 24, 51, 63syl3anc 1199 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ≤ 0 ↔ (((1st𝑟)𝑀(1st𝑠)) ≤ 0 ∧ ((2nd𝑟)𝑁(2nd𝑠)) ≤ 0)))
6561, 62, 643bitrd 213 . . 3 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) = 0 ↔ (((1st𝑟)𝑀(1st𝑠)) ≤ 0 ∧ ((2nd𝑟)𝑁(2nd𝑠)) ≤ 0)))
6655biantrud 300 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1st𝑟)𝑀(1st𝑠)) ≤ 0 ↔ (((1st𝑟)𝑀(1st𝑠)) ≤ 0 ∧ 0 ≤ ((1st𝑟)𝑀(1st𝑠)))))
67 xrletri3 9528 . . . . . 6 ((((1st𝑟)𝑀(1st𝑠)) ∈ ℝ* ∧ 0 ∈ ℝ*) → (((1st𝑟)𝑀(1st𝑠)) = 0 ↔ (((1st𝑟)𝑀(1st𝑠)) ≤ 0 ∧ 0 ≤ ((1st𝑟)𝑀(1st𝑠)))))
6817, 51, 67syl2anc 406 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1st𝑟)𝑀(1st𝑠)) = 0 ↔ (((1st𝑟)𝑀(1st𝑠)) ≤ 0 ∧ 0 ≤ ((1st𝑟)𝑀(1st𝑠)))))
6966, 68bitr4d 190 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1st𝑟)𝑀(1st𝑠)) ≤ 0 ↔ ((1st𝑟)𝑀(1st𝑠)) = 0))
70 xmetge0 12429 . . . . . . 7 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝑟) ∈ 𝑌 ∧ (2nd𝑠) ∈ 𝑌) → 0 ≤ ((2nd𝑟)𝑁(2nd𝑠)))
7118, 20, 22, 70syl3anc 1199 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → 0 ≤ ((2nd𝑟)𝑁(2nd𝑠)))
7271biantrud 300 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((2nd𝑟)𝑁(2nd𝑠)) ≤ 0 ↔ (((2nd𝑟)𝑁(2nd𝑠)) ≤ 0 ∧ 0 ≤ ((2nd𝑟)𝑁(2nd𝑠)))))
73 xrletri3 9528 . . . . . 6 ((((2nd𝑟)𝑁(2nd𝑠)) ∈ ℝ* ∧ 0 ∈ ℝ*) → (((2nd𝑟)𝑁(2nd𝑠)) = 0 ↔ (((2nd𝑟)𝑁(2nd𝑠)) ≤ 0 ∧ 0 ≤ ((2nd𝑟)𝑁(2nd𝑠)))))
7424, 51, 73syl2anc 406 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((2nd𝑟)𝑁(2nd𝑠)) = 0 ↔ (((2nd𝑟)𝑁(2nd𝑠)) ≤ 0 ∧ 0 ≤ ((2nd𝑟)𝑁(2nd𝑠)))))
7572, 74bitr4d 190 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((2nd𝑟)𝑁(2nd𝑠)) ≤ 0 ↔ ((2nd𝑟)𝑁(2nd𝑠)) = 0))
7669, 75anbi12d 462 . . 3 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((((1st𝑟)𝑀(1st𝑠)) ≤ 0 ∧ ((2nd𝑟)𝑁(2nd𝑠)) ≤ 0) ↔ (((1st𝑟)𝑀(1st𝑠)) = 0 ∧ ((2nd𝑟)𝑁(2nd𝑠)) = 0)))
77 xmeteq0 12423 . . . . . 6 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝑟) ∈ 𝑋 ∧ (1st𝑠) ∈ 𝑋) → (((1st𝑟)𝑀(1st𝑠)) = 0 ↔ (1st𝑟) = (1st𝑠)))
7811, 13, 15, 77syl3anc 1199 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1st𝑟)𝑀(1st𝑠)) = 0 ↔ (1st𝑟) = (1st𝑠)))
79 xmeteq0 12423 . . . . . 6 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝑟) ∈ 𝑌 ∧ (2nd𝑠) ∈ 𝑌) → (((2nd𝑟)𝑁(2nd𝑠)) = 0 ↔ (2nd𝑟) = (2nd𝑠)))
8018, 20, 22, 79syl3anc 1199 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((2nd𝑟)𝑁(2nd𝑠)) = 0 ↔ (2nd𝑟) = (2nd𝑠)))
8178, 80anbi12d 462 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((((1st𝑟)𝑀(1st𝑠)) = 0 ∧ ((2nd𝑟)𝑁(2nd𝑠)) = 0) ↔ ((1st𝑟) = (1st𝑠) ∧ (2nd𝑟) = (2nd𝑠))))
82 xpopth 6040 . . . . 5 ((𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌)) → (((1st𝑟) = (1st𝑠) ∧ (2nd𝑟) = (2nd𝑠)) ↔ 𝑟 = 𝑠))
8382adantl 273 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → (((1st𝑟) = (1st𝑠) ∧ (2nd𝑟) = (2nd𝑠)) ↔ 𝑟 = 𝑠))
8481, 83bitrd 187 . . 3 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((((1st𝑟)𝑀(1st𝑠)) = 0 ∧ ((2nd𝑟)𝑁(2nd𝑠)) = 0) ↔ 𝑟 = 𝑠))
8565, 76, 843bitrd 213 . 2 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌))) → ((𝑟𝑃𝑠) = 0 ↔ 𝑟 = 𝑠))
86483adantr3 1125 . . 3 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) = sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ))
87173adantr3 1125 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1st𝑟)𝑀(1st𝑠)) ∈ ℝ*)
881adantr 272 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑀 ∈ (∞Met‘𝑋))
89 simpr3 972 . . . . . . . 8 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑡 ∈ (𝑋 × 𝑌))
90 xp1st 6029 . . . . . . . 8 (𝑡 ∈ (𝑋 × 𝑌) → (1st𝑡) ∈ 𝑋)
9189, 90syl 14 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (1st𝑡) ∈ 𝑋)
92 simpr1 970 . . . . . . . 8 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑟 ∈ (𝑋 × 𝑌))
9392, 12syl 14 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (1st𝑟) ∈ 𝑋)
94 xmetcl 12416 . . . . . . 7 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝑡) ∈ 𝑋 ∧ (1st𝑟) ∈ 𝑋) → ((1st𝑡)𝑀(1st𝑟)) ∈ ℝ*)
9588, 91, 93, 94syl3anc 1199 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1st𝑡)𝑀(1st𝑟)) ∈ ℝ*)
96153adantr3 1125 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (1st𝑠) ∈ 𝑋)
97 xmetcl 12416 . . . . . . 7 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝑡) ∈ 𝑋 ∧ (1st𝑠) ∈ 𝑋) → ((1st𝑡)𝑀(1st𝑠)) ∈ ℝ*)
9888, 91, 96, 97syl3anc 1199 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1st𝑡)𝑀(1st𝑠)) ∈ ℝ*)
9995, 98xaddcld 9607 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (((1st𝑡)𝑀(1st𝑟)) +𝑒 ((1st𝑡)𝑀(1st𝑠))) ∈ ℝ*)
1005adantr 272 . . . . . . . . . 10 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑁 ∈ (∞Met‘𝑌))
101 xp2nd 6030 . . . . . . . . . . 11 (𝑡 ∈ (𝑋 × 𝑌) → (2nd𝑡) ∈ 𝑌)
10289, 101syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (2nd𝑡) ∈ 𝑌)
10392, 19syl 14 . . . . . . . . . 10 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (2nd𝑟) ∈ 𝑌)
104 xmetcl 12416 . . . . . . . . . 10 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝑡) ∈ 𝑌 ∧ (2nd𝑟) ∈ 𝑌) → ((2nd𝑡)𝑁(2nd𝑟)) ∈ ℝ*)
105100, 102, 103, 104syl3anc 1199 . . . . . . . . 9 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2nd𝑡)𝑁(2nd𝑟)) ∈ ℝ*)
106 xrmaxcl 10961 . . . . . . . . 9 ((((1st𝑡)𝑀(1st𝑟)) ∈ ℝ* ∧ ((2nd𝑡)𝑁(2nd𝑟)) ∈ ℝ*) → sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ) ∈ ℝ*)
10795, 105, 106syl2anc 406 . . . . . . . 8 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ) ∈ ℝ*)
108 fveq2 5387 . . . . . . . . . . . 12 (𝑢 = 𝑡 → (1st𝑢) = (1st𝑡))
109 fveq2 5387 . . . . . . . . . . . 12 (𝑣 = 𝑟 → (1st𝑣) = (1st𝑟))
110108, 109oveqan12d 5759 . . . . . . . . . . 11 ((𝑢 = 𝑡𝑣 = 𝑟) → ((1st𝑢)𝑀(1st𝑣)) = ((1st𝑡)𝑀(1st𝑟)))
111 fveq2 5387 . . . . . . . . . . . 12 (𝑢 = 𝑡 → (2nd𝑢) = (2nd𝑡))
112 fveq2 5387 . . . . . . . . . . . 12 (𝑣 = 𝑟 → (2nd𝑣) = (2nd𝑟))
113111, 112oveqan12d 5759 . . . . . . . . . . 11 ((𝑢 = 𝑡𝑣 = 𝑟) → ((2nd𝑢)𝑁(2nd𝑣)) = ((2nd𝑡)𝑁(2nd𝑟)))
114110, 113preq12d 3576 . . . . . . . . . 10 ((𝑢 = 𝑡𝑣 = 𝑟) → {((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))} = {((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))})
115114supeq1d 6840 . . . . . . . . 9 ((𝑢 = 𝑡𝑣 = 𝑟) → sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ) = sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ))
116115, 28ovmpoga 5866 . . . . . . . 8 ((𝑡 ∈ (𝑋 × 𝑌) ∧ 𝑟 ∈ (𝑋 × 𝑌) ∧ sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ) ∈ ℝ*) → (𝑡𝑃𝑟) = sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ))
11789, 92, 107, 116syl3anc 1199 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑟) = sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ))
118117, 107eqeltrd 2192 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑟) ∈ ℝ*)
119 simpr2 971 . . . . . . . 8 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → 𝑠 ∈ (𝑋 × 𝑌))
120223adantr3 1125 . . . . . . . . . 10 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (2nd𝑠) ∈ 𝑌)
121 xmetcl 12416 . . . . . . . . . 10 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝑡) ∈ 𝑌 ∧ (2nd𝑠) ∈ 𝑌) → ((2nd𝑡)𝑁(2nd𝑠)) ∈ ℝ*)
122100, 102, 120, 121syl3anc 1199 . . . . . . . . 9 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2nd𝑡)𝑁(2nd𝑠)) ∈ ℝ*)
123 xrmaxcl 10961 . . . . . . . . 9 ((((1st𝑡)𝑀(1st𝑠)) ∈ ℝ* ∧ ((2nd𝑡)𝑁(2nd𝑠)) ∈ ℝ*) → sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ) ∈ ℝ*)
12498, 122, 123syl2anc 406 . . . . . . . 8 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ) ∈ ℝ*)
125108, 35oveqan12d 5759 . . . . . . . . . . 11 ((𝑢 = 𝑡𝑣 = 𝑠) → ((1st𝑢)𝑀(1st𝑣)) = ((1st𝑡)𝑀(1st𝑠)))
126111, 37oveqan12d 5759 . . . . . . . . . . 11 ((𝑢 = 𝑡𝑣 = 𝑠) → ((2nd𝑢)𝑁(2nd𝑣)) = ((2nd𝑡)𝑁(2nd𝑠)))
127125, 126preq12d 3576 . . . . . . . . . 10 ((𝑢 = 𝑡𝑣 = 𝑠) → {((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))} = {((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))})
128127supeq1d 6840 . . . . . . . . 9 ((𝑢 = 𝑡𝑣 = 𝑠) → sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ) = sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ))
129128, 28ovmpoga 5866 . . . . . . . 8 ((𝑡 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ) ∈ ℝ*) → (𝑡𝑃𝑠) = sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ))
13089, 119, 124, 129syl3anc 1199 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑠) = sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ))
131130, 124eqeltrd 2192 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑡𝑃𝑠) ∈ ℝ*)
132118, 131xaddcld 9607 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)) ∈ ℝ*)
133 xmettri2 12425 . . . . . 6 ((𝑀 ∈ (∞Met‘𝑋) ∧ ((1st𝑡) ∈ 𝑋 ∧ (1st𝑟) ∈ 𝑋 ∧ (1st𝑠) ∈ 𝑋)) → ((1st𝑟)𝑀(1st𝑠)) ≤ (((1st𝑡)𝑀(1st𝑟)) +𝑒 ((1st𝑡)𝑀(1st𝑠))))
13488, 91, 93, 96, 133syl13anc 1201 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1st𝑟)𝑀(1st𝑠)) ≤ (((1st𝑡)𝑀(1st𝑟)) +𝑒 ((1st𝑡)𝑀(1st𝑠))))
135 xrmax1sup 10962 . . . . . . . 8 ((((1st𝑡)𝑀(1st𝑟)) ∈ ℝ* ∧ ((2nd𝑡)𝑁(2nd𝑟)) ∈ ℝ*) → ((1st𝑡)𝑀(1st𝑟)) ≤ sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ))
13695, 105, 135syl2anc 406 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1st𝑡)𝑀(1st𝑟)) ≤ sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ))
137136, 117breqtrrd 3924 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1st𝑡)𝑀(1st𝑟)) ≤ (𝑡𝑃𝑟))
138 xrmax1sup 10962 . . . . . . . 8 ((((1st𝑡)𝑀(1st𝑠)) ∈ ℝ* ∧ ((2nd𝑡)𝑁(2nd𝑠)) ∈ ℝ*) → ((1st𝑡)𝑀(1st𝑠)) ≤ sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ))
13998, 122, 138syl2anc 406 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1st𝑡)𝑀(1st𝑠)) ≤ sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ))
140139, 130breqtrrd 3924 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1st𝑡)𝑀(1st𝑠)) ≤ (𝑡𝑃𝑠))
141 xle2add 9602 . . . . . . 7 (((((1st𝑡)𝑀(1st𝑟)) ∈ ℝ* ∧ ((1st𝑡)𝑀(1st𝑠)) ∈ ℝ*) ∧ ((𝑡𝑃𝑟) ∈ ℝ* ∧ (𝑡𝑃𝑠) ∈ ℝ*)) → ((((1st𝑡)𝑀(1st𝑟)) ≤ (𝑡𝑃𝑟) ∧ ((1st𝑡)𝑀(1st𝑠)) ≤ (𝑡𝑃𝑠)) → (((1st𝑡)𝑀(1st𝑟)) +𝑒 ((1st𝑡)𝑀(1st𝑠))) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠))))
14295, 98, 118, 131, 141syl22anc 1200 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((((1st𝑡)𝑀(1st𝑟)) ≤ (𝑡𝑃𝑟) ∧ ((1st𝑡)𝑀(1st𝑠)) ≤ (𝑡𝑃𝑠)) → (((1st𝑡)𝑀(1st𝑟)) +𝑒 ((1st𝑡)𝑀(1st𝑠))) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠))))
143137, 140, 142mp2and 427 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (((1st𝑡)𝑀(1st𝑟)) +𝑒 ((1st𝑡)𝑀(1st𝑠))) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)))
14487, 99, 132, 134, 143xrletrd 9535 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((1st𝑟)𝑀(1st𝑠)) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)))
145243adantr3 1125 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2nd𝑟)𝑁(2nd𝑠)) ∈ ℝ*)
146105, 122xaddcld 9607 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (((2nd𝑡)𝑁(2nd𝑟)) +𝑒 ((2nd𝑡)𝑁(2nd𝑠))) ∈ ℝ*)
147 xmettri2 12425 . . . . . 6 ((𝑁 ∈ (∞Met‘𝑌) ∧ ((2nd𝑡) ∈ 𝑌 ∧ (2nd𝑟) ∈ 𝑌 ∧ (2nd𝑠) ∈ 𝑌)) → ((2nd𝑟)𝑁(2nd𝑠)) ≤ (((2nd𝑡)𝑁(2nd𝑟)) +𝑒 ((2nd𝑡)𝑁(2nd𝑠))))
148100, 102, 103, 120, 147syl13anc 1201 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2nd𝑟)𝑁(2nd𝑠)) ≤ (((2nd𝑡)𝑁(2nd𝑟)) +𝑒 ((2nd𝑡)𝑁(2nd𝑠))))
149 xrmax2sup 10963 . . . . . . . 8 ((((1st𝑡)𝑀(1st𝑟)) ∈ ℝ* ∧ ((2nd𝑡)𝑁(2nd𝑟)) ∈ ℝ*) → ((2nd𝑡)𝑁(2nd𝑟)) ≤ sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ))
15095, 105, 149syl2anc 406 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2nd𝑡)𝑁(2nd𝑟)) ≤ sup({((1st𝑡)𝑀(1st𝑟)), ((2nd𝑡)𝑁(2nd𝑟))}, ℝ*, < ))
151150, 117breqtrrd 3924 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2nd𝑡)𝑁(2nd𝑟)) ≤ (𝑡𝑃𝑟))
152 xrmax2sup 10963 . . . . . . . 8 ((((1st𝑡)𝑀(1st𝑠)) ∈ ℝ* ∧ ((2nd𝑡)𝑁(2nd𝑠)) ∈ ℝ*) → ((2nd𝑡)𝑁(2nd𝑠)) ≤ sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ))
15398, 122, 152syl2anc 406 . . . . . . 7 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2nd𝑡)𝑁(2nd𝑠)) ≤ sup({((1st𝑡)𝑀(1st𝑠)), ((2nd𝑡)𝑁(2nd𝑠))}, ℝ*, < ))
154153, 130breqtrrd 3924 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2nd𝑡)𝑁(2nd𝑠)) ≤ (𝑡𝑃𝑠))
155 xle2add 9602 . . . . . . 7 (((((2nd𝑡)𝑁(2nd𝑟)) ∈ ℝ* ∧ ((2nd𝑡)𝑁(2nd𝑠)) ∈ ℝ*) ∧ ((𝑡𝑃𝑟) ∈ ℝ* ∧ (𝑡𝑃𝑠) ∈ ℝ*)) → ((((2nd𝑡)𝑁(2nd𝑟)) ≤ (𝑡𝑃𝑟) ∧ ((2nd𝑡)𝑁(2nd𝑠)) ≤ (𝑡𝑃𝑠)) → (((2nd𝑡)𝑁(2nd𝑟)) +𝑒 ((2nd𝑡)𝑁(2nd𝑠))) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠))))
156105, 122, 118, 131, 155syl22anc 1200 . . . . . 6 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((((2nd𝑡)𝑁(2nd𝑟)) ≤ (𝑡𝑃𝑟) ∧ ((2nd𝑡)𝑁(2nd𝑠)) ≤ (𝑡𝑃𝑠)) → (((2nd𝑡)𝑁(2nd𝑟)) +𝑒 ((2nd𝑡)𝑁(2nd𝑠))) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠))))
157151, 154, 156mp2and 427 . . . . 5 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (((2nd𝑡)𝑁(2nd𝑟)) +𝑒 ((2nd𝑡)𝑁(2nd𝑠))) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)))
158145, 146, 132, 148, 157xrletrd 9535 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → ((2nd𝑟)𝑁(2nd𝑠)) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)))
159 xrmaxlesup 10968 . . . . 5 ((((1st𝑟)𝑀(1st𝑠)) ∈ ℝ* ∧ ((2nd𝑟)𝑁(2nd𝑠)) ∈ ℝ* ∧ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)) ∈ ℝ*) → (sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)) ↔ (((1st𝑟)𝑀(1st𝑠)) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)) ∧ ((2nd𝑟)𝑁(2nd𝑠)) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)))))
16087, 145, 132, 159syl3anc 1199 . . . 4 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)) ↔ (((1st𝑟)𝑀(1st𝑠)) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)) ∧ ((2nd𝑟)𝑁(2nd𝑠)) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)))))
161144, 158, 160mpbir2and 911 . . 3 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → sup({((1st𝑟)𝑀(1st𝑠)), ((2nd𝑟)𝑁(2nd𝑠))}, ℝ*, < ) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)))
16286, 161eqbrtrd 3918 . 2 ((𝜑 ∧ (𝑟 ∈ (𝑋 × 𝑌) ∧ 𝑠 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌))) → (𝑟𝑃𝑠) ≤ ((𝑡𝑃𝑟) +𝑒 (𝑡𝑃𝑠)))
16310, 44, 85, 162isxmetd 12411 1 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945   = wceq 1314  wcel 1463  wral 2391  Vcvv 2658  {cpr 3496   class class class wbr 3897   × cxp 4505  wf 5087  cfv 5091  (class class class)co 5740  cmpo 5742  1st c1st 6002  2nd c2nd 6003  supcsup 6835  0cc0 7584  *cxr 7763   < clt 7764  cle 7765   +𝑒 cxad 9497  ∞Metcxmet 12044  MetOpencmopn 12049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-map 6510  df-sup 6837  df-inf 6838  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-xneg 9499  df-xadd 9500  df-seqfrec 10159  df-exp 10233  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-topgen 12036  df-psmet 12051  df-xmet 12052  df-bl 12054  df-mopn 12055  df-top 12060  df-topon 12073  df-bases 12105
This theorem is referenced by:  xmetxpbl  12572  xmettxlem  12573  xmettx  12574  txmetcnp  12582
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