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Theorem xmetxp 13977
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 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))
xmetxp.1 (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
xmetxp.2 (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))
Assertion
Ref Expression
xmetxp (πœ‘ β†’ 𝑃 ∈ (∞Metβ€˜(𝑋 Γ— π‘Œ)))
Distinct variable groups:   𝑒,𝑀,𝑣   𝑒,𝑁,𝑣   𝑒,𝑋,𝑣   𝑒,π‘Œ,𝑣
Allowed substitution hints:   πœ‘(𝑣,𝑒)   𝑃(𝑣,𝑒)

Proof of Theorem xmetxp
Dummy variables π‘Ÿ 𝑠 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmetxp.1 . . . 4 (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
2 eqid 2177 . . . . 5 (MetOpenβ€˜π‘€) = (MetOpenβ€˜π‘€)
32mopnm 13918 . . . 4 (𝑀 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 ∈ (MetOpenβ€˜π‘€))
41, 3syl 14 . . 3 (πœ‘ β†’ 𝑋 ∈ (MetOpenβ€˜π‘€))
5 xmetxp.2 . . . 4 (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))
6 eqid 2177 . . . . 5 (MetOpenβ€˜π‘) = (MetOpenβ€˜π‘)
76mopnm 13918 . . . 4 (𝑁 ∈ (∞Metβ€˜π‘Œ) β†’ π‘Œ ∈ (MetOpenβ€˜π‘))
85, 7syl 14 . . 3 (πœ‘ β†’ π‘Œ ∈ (MetOpenβ€˜π‘))
9 xpexg 4740 . . 3 ((𝑋 ∈ (MetOpenβ€˜π‘€) ∧ π‘Œ ∈ (MetOpenβ€˜π‘)) β†’ (𝑋 Γ— π‘Œ) ∈ V)
104, 8, 9syl2anc 411 . 2 (πœ‘ β†’ (𝑋 Γ— π‘Œ) ∈ V)
111adantr 276 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
12 xp1st 6165 . . . . . . 7 (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) β†’ (1st β€˜π‘Ÿ) ∈ 𝑋)
1312ad2antrl 490 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (1st β€˜π‘Ÿ) ∈ 𝑋)
14 xp1st 6165 . . . . . . 7 (𝑠 ∈ (𝑋 Γ— π‘Œ) β†’ (1st β€˜π‘ ) ∈ 𝑋)
1514ad2antll 491 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (1st β€˜π‘ ) ∈ 𝑋)
16 xmetcl 13822 . . . . . 6 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜π‘Ÿ) ∈ 𝑋 ∧ (1st β€˜π‘ ) ∈ 𝑋) β†’ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ∈ ℝ*)
1711, 13, 15, 16syl3anc 1238 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ∈ ℝ*)
185adantr 276 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))
19 xp2nd 6166 . . . . . . 7 (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) β†’ (2nd β€˜π‘Ÿ) ∈ π‘Œ)
2019ad2antrl 490 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (2nd β€˜π‘Ÿ) ∈ π‘Œ)
21 xp2nd 6166 . . . . . . 7 (𝑠 ∈ (𝑋 Γ— π‘Œ) β†’ (2nd β€˜π‘ ) ∈ π‘Œ)
2221ad2antll 491 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (2nd β€˜π‘ ) ∈ π‘Œ)
23 xmetcl 13822 . . . . . 6 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ (2nd β€˜π‘Ÿ) ∈ π‘Œ ∧ (2nd β€˜π‘ ) ∈ π‘Œ) β†’ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ∈ ℝ*)
2418, 20, 22, 23syl3anc 1238 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ∈ ℝ*)
25 xrmaxcl 11259 . . . . 5 ((((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ∈ ℝ* ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ∈ ℝ*) β†’ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ∈ ℝ*)
2617, 24, 25syl2anc 411 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ∈ ℝ*)
2726ralrimivva 2559 . . 3 (πœ‘ β†’ βˆ€π‘Ÿ ∈ (𝑋 Γ— π‘Œ)βˆ€π‘  ∈ (𝑋 Γ— π‘Œ)sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ∈ ℝ*)
28 xmetxp.p . . . . 5 𝑃 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))
29 fveq2 5515 . . . . . . . . 9 (𝑒 = π‘Ÿ β†’ (1st β€˜π‘’) = (1st β€˜π‘Ÿ))
3029oveq1d 5889 . . . . . . . 8 (𝑒 = π‘Ÿ β†’ ((1st β€˜π‘’)𝑀(1st β€˜π‘£)) = ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘£)))
31 fveq2 5515 . . . . . . . . 9 (𝑒 = π‘Ÿ β†’ (2nd β€˜π‘’) = (2nd β€˜π‘Ÿ))
3231oveq1d 5889 . . . . . . . 8 (𝑒 = π‘Ÿ β†’ ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£)) = ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘£)))
3330, 32preq12d 3677 . . . . . . 7 (𝑒 = π‘Ÿ β†’ {((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))} = {((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘£))})
3433supeq1d 6985 . . . . . 6 (𝑒 = π‘Ÿ β†’ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ) = sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘£))}, ℝ*, < ))
35 fveq2 5515 . . . . . . . . 9 (𝑣 = 𝑠 β†’ (1st β€˜π‘£) = (1st β€˜π‘ ))
3635oveq2d 5890 . . . . . . . 8 (𝑣 = 𝑠 β†’ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘£)) = ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )))
37 fveq2 5515 . . . . . . . . 9 (𝑣 = 𝑠 β†’ (2nd β€˜π‘£) = (2nd β€˜π‘ ))
3837oveq2d 5890 . . . . . . . 8 (𝑣 = 𝑠 β†’ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘£)) = ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )))
3936, 38preq12d 3677 . . . . . . 7 (𝑣 = 𝑠 β†’ {((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘£))} = {((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))})
4039supeq1d 6985 . . . . . 6 (𝑣 = 𝑠 β†’ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘£))}, ℝ*, < ) = sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
4134, 40cbvmpov 5954 . . . . 5 (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < )) = (π‘Ÿ ∈ (𝑋 Γ— π‘Œ), 𝑠 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
4228, 41eqtri 2198 . . . 4 𝑃 = (π‘Ÿ ∈ (𝑋 Γ— π‘Œ), 𝑠 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
4342fmpo 6201 . . 3 (βˆ€π‘Ÿ ∈ (𝑋 Γ— π‘Œ)βˆ€π‘  ∈ (𝑋 Γ— π‘Œ)sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ∈ ℝ* ↔ 𝑃:((𝑋 Γ— π‘Œ) Γ— (𝑋 Γ— π‘Œ))βŸΆβ„*)
4427, 43sylib 122 . 2 (πœ‘ β†’ 𝑃:((𝑋 Γ— π‘Œ) Γ— (𝑋 Γ— π‘Œ))βŸΆβ„*)
45 simprl 529 . . . . . . . 8 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ π‘Ÿ ∈ (𝑋 Γ— π‘Œ))
46 simprr 531 . . . . . . . 8 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ 𝑠 ∈ (𝑋 Γ— π‘Œ))
4734, 40, 28ovmpog 6008 . . . . . . . 8 ((π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ∈ ℝ*) β†’ (π‘Ÿπ‘ƒπ‘ ) = sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
4845, 46, 26, 47syl3anc 1238 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (π‘Ÿπ‘ƒπ‘ ) = sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
4948, 26eqeltrd 2254 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (π‘Ÿπ‘ƒπ‘ ) ∈ ℝ*)
50 0xr 8003 . . . . . . 7 0 ∈ ℝ*
5150a1i 9 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ 0 ∈ ℝ*)
52 xrletri3 9803 . . . . . 6 (((π‘Ÿπ‘ƒπ‘ ) ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ ((π‘Ÿπ‘ƒπ‘ ) = 0 ↔ ((π‘Ÿπ‘ƒπ‘ ) ≀ 0 ∧ 0 ≀ (π‘Ÿπ‘ƒπ‘ ))))
5349, 51, 52syl2anc 411 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((π‘Ÿπ‘ƒπ‘ ) = 0 ↔ ((π‘Ÿπ‘ƒπ‘ ) ≀ 0 ∧ 0 ≀ (π‘Ÿπ‘ƒπ‘ ))))
54 xmetge0 13835 . . . . . . . . 9 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜π‘Ÿ) ∈ 𝑋 ∧ (1st β€˜π‘ ) ∈ 𝑋) β†’ 0 ≀ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )))
5511, 13, 15, 54syl3anc 1238 . . . . . . . 8 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ 0 ≀ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )))
56 xrmax1sup 11260 . . . . . . . . 9 ((((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ∈ ℝ* ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ∈ ℝ*) β†’ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
5717, 24, 56syl2anc 411 . . . . . . . 8 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
5851, 17, 26, 55, 57xrletrd 9811 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ 0 ≀ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
5958, 48breqtrrd 4031 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ 0 ≀ (π‘Ÿπ‘ƒπ‘ ))
6059biantrud 304 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((π‘Ÿπ‘ƒπ‘ ) ≀ 0 ↔ ((π‘Ÿπ‘ƒπ‘ ) ≀ 0 ∧ 0 ≀ (π‘Ÿπ‘ƒπ‘ ))))
6153, 60bitr4d 191 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((π‘Ÿπ‘ƒπ‘ ) = 0 ↔ (π‘Ÿπ‘ƒπ‘ ) ≀ 0))
6248breq1d 4013 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((π‘Ÿπ‘ƒπ‘ ) ≀ 0 ↔ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ≀ 0))
63 xrmaxlesup 11266 . . . . 5 ((((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ∈ ℝ* ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ (sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ≀ 0 ↔ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ 0 ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ 0)))
6417, 24, 51, 63syl3anc 1238 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ≀ 0 ↔ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ 0 ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ 0)))
6561, 62, 643bitrd 214 . . 3 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((π‘Ÿπ‘ƒπ‘ ) = 0 ↔ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ 0 ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ 0)))
6655biantrud 304 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ 0 ↔ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ 0 ∧ 0 ≀ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )))))
67 xrletri3 9803 . . . . . 6 ((((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) = 0 ↔ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ 0 ∧ 0 ≀ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )))))
6817, 51, 67syl2anc 411 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) = 0 ↔ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ 0 ∧ 0 ≀ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )))))
6966, 68bitr4d 191 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ 0 ↔ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) = 0))
70 xmetge0 13835 . . . . . . 7 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ (2nd β€˜π‘Ÿ) ∈ π‘Œ ∧ (2nd β€˜π‘ ) ∈ π‘Œ) β†’ 0 ≀ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )))
7118, 20, 22, 70syl3anc 1238 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ 0 ≀ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )))
7271biantrud 304 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ 0 ↔ (((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ 0 ∧ 0 ≀ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )))))
73 xrletri3 9803 . . . . . 6 ((((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ (((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) = 0 ↔ (((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ 0 ∧ 0 ≀ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )))))
7424, 51, 73syl2anc 411 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) = 0 ↔ (((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ 0 ∧ 0 ≀ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )))))
7572, 74bitr4d 191 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ 0 ↔ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) = 0))
7669, 75anbi12d 473 . . 3 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ 0 ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ 0) ↔ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) = 0 ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) = 0)))
77 xmeteq0 13829 . . . . . 6 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜π‘Ÿ) ∈ 𝑋 ∧ (1st β€˜π‘ ) ∈ 𝑋) β†’ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) = 0 ↔ (1st β€˜π‘Ÿ) = (1st β€˜π‘ )))
7811, 13, 15, 77syl3anc 1238 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) = 0 ↔ (1st β€˜π‘Ÿ) = (1st β€˜π‘ )))
79 xmeteq0 13829 . . . . . 6 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ (2nd β€˜π‘Ÿ) ∈ π‘Œ ∧ (2nd β€˜π‘ ) ∈ π‘Œ) β†’ (((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) = 0 ↔ (2nd β€˜π‘Ÿ) = (2nd β€˜π‘ )))
8018, 20, 22, 79syl3anc 1238 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) = 0 ↔ (2nd β€˜π‘Ÿ) = (2nd β€˜π‘ )))
8178, 80anbi12d 473 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) = 0 ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) = 0) ↔ ((1st β€˜π‘Ÿ) = (1st β€˜π‘ ) ∧ (2nd β€˜π‘Ÿ) = (2nd β€˜π‘ ))))
82 xpopth 6176 . . . . 5 ((π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ)) β†’ (((1st β€˜π‘Ÿ) = (1st β€˜π‘ ) ∧ (2nd β€˜π‘Ÿ) = (2nd β€˜π‘ )) ↔ π‘Ÿ = 𝑠))
8382adantl 277 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ (((1st β€˜π‘Ÿ) = (1st β€˜π‘ ) ∧ (2nd β€˜π‘Ÿ) = (2nd β€˜π‘ )) ↔ π‘Ÿ = 𝑠))
8481, 83bitrd 188 . . 3 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) = 0 ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) = 0) ↔ π‘Ÿ = 𝑠))
8565, 76, 843bitrd 214 . 2 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ))) β†’ ((π‘Ÿπ‘ƒπ‘ ) = 0 ↔ π‘Ÿ = 𝑠))
86483adantr3 1158 . . 3 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (π‘Ÿπ‘ƒπ‘ ) = sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
87173adantr3 1158 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ∈ ℝ*)
881adantr 276 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
89 simpr3 1005 . . . . . . . 8 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ 𝑑 ∈ (𝑋 Γ— π‘Œ))
90 xp1st 6165 . . . . . . . 8 (𝑑 ∈ (𝑋 Γ— π‘Œ) β†’ (1st β€˜π‘‘) ∈ 𝑋)
9189, 90syl 14 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (1st β€˜π‘‘) ∈ 𝑋)
92 simpr1 1003 . . . . . . . 8 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ π‘Ÿ ∈ (𝑋 Γ— π‘Œ))
9392, 12syl 14 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (1st β€˜π‘Ÿ) ∈ 𝑋)
94 xmetcl 13822 . . . . . . 7 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜π‘‘) ∈ 𝑋 ∧ (1st β€˜π‘Ÿ) ∈ 𝑋) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ∈ ℝ*)
9588, 91, 93, 94syl3anc 1238 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ∈ ℝ*)
96153adantr3 1158 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (1st β€˜π‘ ) ∈ 𝑋)
97 xmetcl 13822 . . . . . . 7 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜π‘‘) ∈ 𝑋 ∧ (1st β€˜π‘ ) ∈ 𝑋) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ∈ ℝ*)
9888, 91, 96, 97syl3anc 1238 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ∈ ℝ*)
9995, 98xaddcld 9883 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) +𝑒 ((1st β€˜π‘‘)𝑀(1st β€˜π‘ ))) ∈ ℝ*)
1005adantr 276 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))
101 xp2nd 6166 . . . . . . . . . . 11 (𝑑 ∈ (𝑋 Γ— π‘Œ) β†’ (2nd β€˜π‘‘) ∈ π‘Œ)
10289, 101syl 14 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (2nd β€˜π‘‘) ∈ π‘Œ)
10392, 19syl 14 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (2nd β€˜π‘Ÿ) ∈ π‘Œ)
104 xmetcl 13822 . . . . . . . . . 10 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ (2nd β€˜π‘‘) ∈ π‘Œ ∧ (2nd β€˜π‘Ÿ) ∈ π‘Œ) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ∈ ℝ*)
105100, 102, 103, 104syl3anc 1238 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ∈ ℝ*)
106 xrmaxcl 11259 . . . . . . . . 9 ((((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ∈ ℝ* ∧ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ∈ ℝ*) β†’ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ) ∈ ℝ*)
10795, 105, 106syl2anc 411 . . . . . . . 8 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ) ∈ ℝ*)
108 fveq2 5515 . . . . . . . . . . . 12 (𝑒 = 𝑑 β†’ (1st β€˜π‘’) = (1st β€˜π‘‘))
109 fveq2 5515 . . . . . . . . . . . 12 (𝑣 = π‘Ÿ β†’ (1st β€˜π‘£) = (1st β€˜π‘Ÿ))
110108, 109oveqan12d 5893 . . . . . . . . . . 11 ((𝑒 = 𝑑 ∧ 𝑣 = π‘Ÿ) β†’ ((1st β€˜π‘’)𝑀(1st β€˜π‘£)) = ((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)))
111 fveq2 5515 . . . . . . . . . . . 12 (𝑒 = 𝑑 β†’ (2nd β€˜π‘’) = (2nd β€˜π‘‘))
112 fveq2 5515 . . . . . . . . . . . 12 (𝑣 = π‘Ÿ β†’ (2nd β€˜π‘£) = (2nd β€˜π‘Ÿ))
113111, 112oveqan12d 5893 . . . . . . . . . . 11 ((𝑒 = 𝑑 ∧ 𝑣 = π‘Ÿ) β†’ ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£)) = ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)))
114110, 113preq12d 3677 . . . . . . . . . 10 ((𝑒 = 𝑑 ∧ 𝑣 = π‘Ÿ) β†’ {((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))} = {((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))})
115114supeq1d 6985 . . . . . . . . 9 ((𝑒 = 𝑑 ∧ 𝑣 = π‘Ÿ) β†’ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ) = sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ))
116115, 28ovmpoga 6003 . . . . . . . 8 ((𝑑 ∈ (𝑋 Γ— π‘Œ) ∧ π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ) ∈ ℝ*) β†’ (π‘‘π‘ƒπ‘Ÿ) = sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ))
11789, 92, 107, 116syl3anc 1238 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (π‘‘π‘ƒπ‘Ÿ) = sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ))
118117, 107eqeltrd 2254 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (π‘‘π‘ƒπ‘Ÿ) ∈ ℝ*)
119 simpr2 1004 . . . . . . . 8 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ 𝑠 ∈ (𝑋 Γ— π‘Œ))
120223adantr3 1158 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (2nd β€˜π‘ ) ∈ π‘Œ)
121 xmetcl 13822 . . . . . . . . . 10 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ (2nd β€˜π‘‘) ∈ π‘Œ ∧ (2nd β€˜π‘ ) ∈ π‘Œ) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ∈ ℝ*)
122100, 102, 120, 121syl3anc 1238 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ∈ ℝ*)
123 xrmaxcl 11259 . . . . . . . . 9 ((((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ∈ ℝ* ∧ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ∈ ℝ*) β†’ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ∈ ℝ*)
12498, 122, 123syl2anc 411 . . . . . . . 8 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ∈ ℝ*)
125108, 35oveqan12d 5893 . . . . . . . . . . 11 ((𝑒 = 𝑑 ∧ 𝑣 = 𝑠) β†’ ((1st β€˜π‘’)𝑀(1st β€˜π‘£)) = ((1st β€˜π‘‘)𝑀(1st β€˜π‘ )))
126111, 37oveqan12d 5893 . . . . . . . . . . 11 ((𝑒 = 𝑑 ∧ 𝑣 = 𝑠) β†’ ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£)) = ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )))
127125, 126preq12d 3677 . . . . . . . . . 10 ((𝑒 = 𝑑 ∧ 𝑣 = 𝑠) β†’ {((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))} = {((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))})
128127supeq1d 6985 . . . . . . . . 9 ((𝑒 = 𝑑 ∧ 𝑣 = 𝑠) β†’ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ) = sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
129128, 28ovmpoga 6003 . . . . . . . 8 ((𝑑 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ∈ ℝ*) β†’ (𝑑𝑃𝑠) = sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
13089, 119, 124, 129syl3anc 1238 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (𝑑𝑃𝑠) = sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
131130, 124eqeltrd 2254 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (𝑑𝑃𝑠) ∈ ℝ*)
132118, 131xaddcld 9883 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)) ∈ ℝ*)
133 xmettri2 13831 . . . . . 6 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ ((1st β€˜π‘‘) ∈ 𝑋 ∧ (1st β€˜π‘Ÿ) ∈ 𝑋 ∧ (1st β€˜π‘ ) ∈ 𝑋)) β†’ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ (((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) +𝑒 ((1st β€˜π‘‘)𝑀(1st β€˜π‘ ))))
13488, 91, 93, 96, 133syl13anc 1240 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ (((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) +𝑒 ((1st β€˜π‘‘)𝑀(1st β€˜π‘ ))))
135 xrmax1sup 11260 . . . . . . . 8 ((((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ∈ ℝ* ∧ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ∈ ℝ*) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ≀ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ))
13695, 105, 135syl2anc 411 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ≀ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ))
137136, 117breqtrrd 4031 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ≀ (π‘‘π‘ƒπ‘Ÿ))
138 xrmax1sup 11260 . . . . . . . 8 ((((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ∈ ℝ* ∧ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ∈ ℝ*) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ≀ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
13998, 122, 138syl2anc 411 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ≀ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
140139, 130breqtrrd 4031 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ≀ (𝑑𝑃𝑠))
141 xle2add 9878 . . . . . . 7 (((((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ∈ ℝ* ∧ ((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ∈ ℝ*) ∧ ((π‘‘π‘ƒπ‘Ÿ) ∈ ℝ* ∧ (𝑑𝑃𝑠) ∈ ℝ*)) β†’ ((((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ≀ (π‘‘π‘ƒπ‘Ÿ) ∧ ((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ≀ (𝑑𝑃𝑠)) β†’ (((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) +𝑒 ((1st β€˜π‘‘)𝑀(1st β€˜π‘ ))) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠))))
14295, 98, 118, 131, 141syl22anc 1239 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ≀ (π‘‘π‘ƒπ‘Ÿ) ∧ ((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ≀ (𝑑𝑃𝑠)) β†’ (((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) +𝑒 ((1st β€˜π‘‘)𝑀(1st β€˜π‘ ))) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠))))
143137, 140, 142mp2and 433 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) +𝑒 ((1st β€˜π‘‘)𝑀(1st β€˜π‘ ))) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)))
14487, 99, 132, 134, 143xrletrd 9811 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)))
145243adantr3 1158 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ∈ ℝ*)
146105, 122xaddcld 9883 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) +𝑒 ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))) ∈ ℝ*)
147 xmettri2 13831 . . . . . 6 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ ((2nd β€˜π‘‘) ∈ π‘Œ ∧ (2nd β€˜π‘Ÿ) ∈ π‘Œ ∧ (2nd β€˜π‘ ) ∈ π‘Œ)) β†’ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ (((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) +𝑒 ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))))
148100, 102, 103, 120, 147syl13anc 1240 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ (((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) +𝑒 ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))))
149 xrmax2sup 11261 . . . . . . . 8 ((((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)) ∈ ℝ* ∧ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ∈ ℝ*) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ≀ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ))
15095, 105, 149syl2anc 411 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ≀ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘Ÿ)), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ))}, ℝ*, < ))
151150, 117breqtrrd 4031 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ≀ (π‘‘π‘ƒπ‘Ÿ))
152 xrmax2sup 11261 . . . . . . . 8 ((((1st β€˜π‘‘)𝑀(1st β€˜π‘ )) ∈ ℝ* ∧ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ∈ ℝ*) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ≀ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
15398, 122, 152syl2anc 411 . . . . . . 7 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ≀ sup({((1st β€˜π‘‘)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))}, ℝ*, < ))
154153, 130breqtrrd 4031 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ≀ (𝑑𝑃𝑠))
155 xle2add 9878 . . . . . . 7 (((((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ∈ ℝ* ∧ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ∈ ℝ*) ∧ ((π‘‘π‘ƒπ‘Ÿ) ∈ ℝ* ∧ (𝑑𝑃𝑠) ∈ ℝ*)) β†’ ((((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ≀ (π‘‘π‘ƒπ‘Ÿ) ∧ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ≀ (𝑑𝑃𝑠)) β†’ (((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) +𝑒 ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠))))
156105, 122, 118, 131, 155syl22anc 1239 . . . . . 6 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) ≀ (π‘‘π‘ƒπ‘Ÿ) ∧ ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ )) ≀ (𝑑𝑃𝑠)) β†’ (((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) +𝑒 ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠))))
157151, 154, 156mp2and 433 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (((2nd β€˜π‘‘)𝑁(2nd β€˜π‘Ÿ)) +𝑒 ((2nd β€˜π‘‘)𝑁(2nd β€˜π‘ ))) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)))
158145, 146, 132, 148, 157xrletrd 9811 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)))
159 xrmaxlesup 11266 . . . . 5 ((((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ∈ ℝ* ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ∈ ℝ* ∧ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)) ∈ ℝ*) β†’ (sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)) ↔ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)) ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)))))
16087, 145, 132, 159syl3anc 1238 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)) ↔ (((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)) ∧ ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ )) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)))))
161144, 158, 160mpbir2and 944 . . 3 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ sup({((1st β€˜π‘Ÿ)𝑀(1st β€˜π‘ )), ((2nd β€˜π‘Ÿ)𝑁(2nd β€˜π‘ ))}, ℝ*, < ) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)))
16286, 161eqbrtrd 4025 . 2 ((πœ‘ ∧ (π‘Ÿ ∈ (𝑋 Γ— π‘Œ) ∧ 𝑠 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑑 ∈ (𝑋 Γ— π‘Œ))) β†’ (π‘Ÿπ‘ƒπ‘ ) ≀ ((π‘‘π‘ƒπ‘Ÿ) +𝑒 (𝑑𝑃𝑠)))
16310, 44, 85, 162isxmetd 13817 1 (πœ‘ β†’ 𝑃 ∈ (∞Metβ€˜(𝑋 Γ— π‘Œ)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  Vcvv 2737  {cpr 3593   class class class wbr 4003   Γ— cxp 4624  βŸΆwf 5212  β€˜cfv 5216  (class class class)co 5874   ∈ cmpo 5876  1st c1st 6138  2nd c2nd 6139  supcsup 6980  0cc0 7810  β„*cxr 7990   < clt 7991   ≀ cle 7992   +𝑒 cxad 9769  βˆžMetcxmet 13410  MetOpencmopn 13415
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-map 6649  df-sup 6982  df-inf 6983  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-xneg 9771  df-xadd 9772  df-seqfrec 10445  df-exp 10519  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-topgen 12708  df-psmet 13417  df-xmet 13418  df-bl 13420  df-mopn 13421  df-top 13468  df-topon 13481  df-bases 13513
This theorem is referenced by:  xmetxpbl  13978  xmettxlem  13979  xmettx  13980  txmetcnp  13988
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