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Theorem prdsxmetlem 23643
Description: The product metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
prdsdsf.y π‘Œ = (𝑆Xs(π‘₯ ∈ 𝐼 ↦ 𝑅))
prdsdsf.b 𝐡 = (Baseβ€˜π‘Œ)
prdsdsf.v 𝑉 = (Baseβ€˜π‘…)
prdsdsf.e 𝐸 = ((distβ€˜π‘…) β†Ύ (𝑉 Γ— 𝑉))
prdsdsf.d 𝐷 = (distβ€˜π‘Œ)
prdsdsf.s (πœ‘ β†’ 𝑆 ∈ π‘Š)
prdsdsf.i (πœ‘ β†’ 𝐼 ∈ 𝑋)
prdsdsf.r ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝑅 ∈ 𝑍)
prdsdsf.m ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝐸 ∈ (∞Metβ€˜π‘‰))
Assertion
Ref Expression
prdsxmetlem (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π΅))
Distinct variable groups:   π‘₯,𝐼   πœ‘,π‘₯   π‘₯,𝐡   π‘₯,𝐷
Allowed substitution hints:   𝑅(π‘₯)   𝑆(π‘₯)   𝐸(π‘₯)   𝑉(π‘₯)   π‘Š(π‘₯)   𝑋(π‘₯)   π‘Œ(π‘₯)   𝑍(π‘₯)

Proof of Theorem prdsxmetlem
Dummy variables 𝑓 𝑔 β„Ž 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsdsf.b . . . 4 𝐡 = (Baseβ€˜π‘Œ)
21fvexi 6851 . . 3 𝐡 ∈ V
32a1i 11 . 2 (πœ‘ β†’ 𝐡 ∈ V)
4 prdsdsf.y . . . 4 π‘Œ = (𝑆Xs(π‘₯ ∈ 𝐼 ↦ 𝑅))
5 prdsdsf.v . . . 4 𝑉 = (Baseβ€˜π‘…)
6 prdsdsf.e . . . 4 𝐸 = ((distβ€˜π‘…) β†Ύ (𝑉 Γ— 𝑉))
7 prdsdsf.d . . . 4 𝐷 = (distβ€˜π‘Œ)
8 prdsdsf.s . . . 4 (πœ‘ β†’ 𝑆 ∈ π‘Š)
9 prdsdsf.i . . . 4 (πœ‘ β†’ 𝐼 ∈ 𝑋)
10 prdsdsf.r . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝑅 ∈ 𝑍)
11 prdsdsf.m . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝐸 ∈ (∞Metβ€˜π‘‰))
124, 1, 5, 6, 7, 8, 9, 10, 11prdsdsf 23642 . . 3 (πœ‘ β†’ 𝐷:(𝐡 Γ— 𝐡)⟢(0[,]+∞))
13 iccssxr 13275 . . 3 (0[,]+∞) βŠ† ℝ*
14 fss 6680 . . 3 ((𝐷:(𝐡 Γ— 𝐡)⟢(0[,]+∞) ∧ (0[,]+∞) βŠ† ℝ*) β†’ 𝐷:(𝐡 Γ— 𝐡)βŸΆβ„*)
1512, 13, 14sylancl 586 . 2 (πœ‘ β†’ 𝐷:(𝐡 Γ— 𝐡)βŸΆβ„*)
1612fovcdmda 7517 . . 3 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑓𝐷𝑔) ∈ (0[,]+∞))
17 elxrge0 13302 . . . 4 ((𝑓𝐷𝑔) ∈ (0[,]+∞) ↔ ((𝑓𝐷𝑔) ∈ ℝ* ∧ 0 ≀ (𝑓𝐷𝑔)))
1817simprbi 497 . . 3 ((𝑓𝐷𝑔) ∈ (0[,]+∞) β†’ 0 ≀ (𝑓𝐷𝑔))
1916, 18syl 17 . 2 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 0 ≀ (𝑓𝐷𝑔))
208adantr 481 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑆 ∈ π‘Š)
219adantr 481 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝐼 ∈ 𝑋)
2210ralrimiva 3141 . . . . . 6 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐼 𝑅 ∈ 𝑍)
2322adantr 481 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 𝑅 ∈ 𝑍)
24 simprl 769 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑓 ∈ 𝐡)
25 simprr 771 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑔 ∈ 𝐡)
264, 1, 20, 21, 23, 24, 25, 5, 6, 7prdsdsval3 17301 . . . 4 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑓𝐷𝑔) = sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
2726breq1d 5113 . . 3 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ ((𝑓𝐷𝑔) ≀ 0 ↔ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ) ≀ 0))
2811adantlr 713 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ 𝐸 ∈ (∞Metβ€˜π‘‰))
294, 1, 20, 21, 23, 5, 24prdsbascl 17299 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) ∈ 𝑉)
3029r19.21bi 3232 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ (π‘“β€˜π‘₯) ∈ 𝑉)
314, 1, 20, 21, 23, 5, 25prdsbascl 17299 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘”β€˜π‘₯) ∈ 𝑉)
3231r19.21bi 3232 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ (π‘”β€˜π‘₯) ∈ 𝑉)
33 xmetcl 23606 . . . . . . . 8 ((𝐸 ∈ (∞Metβ€˜π‘‰) ∧ (π‘“β€˜π‘₯) ∈ 𝑉 ∧ (π‘”β€˜π‘₯) ∈ 𝑉) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ*)
3428, 30, 32, 33syl3anc 1371 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ*)
3534fmpttd 7057 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))):πΌβŸΆβ„*)
3635frnd 6671 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βŠ† ℝ*)
37 0xr 11135 . . . . . . 7 0 ∈ ℝ*
3837a1i 11 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 0 ∈ ℝ*)
3938snssd 4767 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ {0} βŠ† ℝ*)
4036, 39unssd 4144 . . . 4 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ*)
41 supxrleub 13173 . . . 4 (((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ* ∧ 0 ∈ ℝ*) β†’ (sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ) ≀ 0 ↔ βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ 0))
4240, 37, 41sylancl 586 . . 3 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ) ≀ 0 ↔ βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ 0))
43 0le0 12187 . . . . . . 7 0 ≀ 0
44 c0ex 11082 . . . . . . . 8 0 ∈ V
45 breq1 5106 . . . . . . . 8 (𝑧 = 0 β†’ (𝑧 ≀ 0 ↔ 0 ≀ 0))
4644, 45ralsn 4640 . . . . . . 7 (βˆ€π‘§ ∈ {0}𝑧 ≀ 0 ↔ 0 ≀ 0)
4743, 46mpbir 230 . . . . . 6 βˆ€π‘§ ∈ {0}𝑧 ≀ 0
48 ralunb 4149 . . . . . 6 (βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ 0 ↔ (βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))𝑧 ≀ 0 ∧ βˆ€π‘§ ∈ {0}𝑧 ≀ 0))
4947, 48mpbiran2 708 . . . . 5 (βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ 0 ↔ βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))𝑧 ≀ 0)
50 ovex 7382 . . . . . . 7 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ V
5150rgenw 3066 . . . . . 6 βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ V
52 eqid 2737 . . . . . . 7 (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
53 breq1 5106 . . . . . . 7 (𝑧 = ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) β†’ (𝑧 ≀ 0 ↔ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0))
5452, 53ralrnmptw 7038 . . . . . 6 (βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ V β†’ (βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))𝑧 ≀ 0 ↔ βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0))
5551, 54ax-mp 5 . . . . 5 (βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))𝑧 ≀ 0 ↔ βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0)
5649, 55bitri 274 . . . 4 (βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ 0 ↔ βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0)
57 xmetge0 23619 . . . . . . . . 9 ((𝐸 ∈ (∞Metβ€˜π‘‰) ∧ (π‘“β€˜π‘₯) ∈ 𝑉 ∧ (π‘”β€˜π‘₯) ∈ 𝑉) β†’ 0 ≀ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
5828, 30, 32, 57syl3anc 1371 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ 0 ≀ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
5958biantrud 532 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ (((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0 ↔ (((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0 ∧ 0 ≀ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))))
60 xrletri3 13001 . . . . . . . 8 ((((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ (((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) = 0 ↔ (((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0 ∧ 0 ≀ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))))
6134, 37, 60sylancl 586 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ (((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) = 0 ↔ (((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0 ∧ 0 ≀ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))))
62 xmeteq0 23613 . . . . . . . 8 ((𝐸 ∈ (∞Metβ€˜π‘‰) ∧ (π‘“β€˜π‘₯) ∈ 𝑉 ∧ (π‘”β€˜π‘₯) ∈ 𝑉) β†’ (((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) = 0 ↔ (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯)))
6328, 30, 32, 62syl3anc 1371 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ (((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) = 0 ↔ (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯)))
6459, 61, 633bitr2d 306 . . . . . 6 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ (((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0 ↔ (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯)))
6564ralbidva 3170 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0 ↔ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯)))
66 eqid 2737 . . . . . . . . . 10 (π‘₯ ∈ 𝐼 ↦ 𝑅) = (π‘₯ ∈ 𝐼 ↦ 𝑅)
6766fnmpt 6636 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝐼 𝑅 ∈ 𝑍 β†’ (π‘₯ ∈ 𝐼 ↦ 𝑅) Fn 𝐼)
6822, 67syl 17 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ 𝐼 ↦ 𝑅) Fn 𝐼)
6968adantr 481 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (π‘₯ ∈ 𝐼 ↦ 𝑅) Fn 𝐼)
704, 1, 20, 21, 69, 24prdsbasfn 17287 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑓 Fn 𝐼)
714, 1, 20, 21, 69, 25prdsbasfn 17287 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑔 Fn 𝐼)
72 eqfnfv 6977 . . . . . 6 ((𝑓 Fn 𝐼 ∧ 𝑔 Fn 𝐼) β†’ (𝑓 = 𝑔 ↔ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯)))
7370, 71, 72syl2anc 584 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑓 = 𝑔 ↔ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯)))
7465, 73bitr4d 281 . . . 4 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0 ↔ 𝑓 = 𝑔))
7556, 74bitrid 282 . . 3 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ 0 ↔ 𝑓 = 𝑔))
7627, 42, 753bitrd 304 . 2 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ ((𝑓𝐷𝑔) ≀ 0 ↔ 𝑓 = 𝑔))
77263adantr3 1171 . . . 4 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡)) β†’ (𝑓𝐷𝑔) = sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
78773adant3 1132 . . 3 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (𝑓𝐷𝑔) = sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
79113ad2antl1 1185 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ 𝐸 ∈ (∞Metβ€˜π‘‰))
80293adantr3 1171 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) ∈ 𝑉)
81803adant3 1132 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) ∈ 𝑉)
8281r19.21bi 3232 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (π‘“β€˜π‘₯) ∈ 𝑉)
83313adantr3 1171 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘”β€˜π‘₯) ∈ 𝑉)
84833adant3 1132 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘”β€˜π‘₯) ∈ 𝑉)
8584r19.21bi 3232 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (π‘”β€˜π‘₯) ∈ 𝑉)
8679, 82, 85, 33syl3anc 1371 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ*)
8783ad2ant1 1133 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 𝑆 ∈ π‘Š)
8893ad2ant1 1133 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 𝐼 ∈ 𝑋)
89223ad2ant1 1133 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ βˆ€π‘₯ ∈ 𝐼 𝑅 ∈ 𝑍)
90 simp23 1208 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ β„Ž ∈ 𝐡)
914, 1, 87, 88, 89, 5, 90prdsbascl 17299 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ βˆ€π‘₯ ∈ 𝐼 (β„Žβ€˜π‘₯) ∈ 𝑉)
9291r19.21bi 3232 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (β„Žβ€˜π‘₯) ∈ 𝑉)
93 xmetcl 23606 . . . . . . . . . . . 12 ((𝐸 ∈ (∞Metβ€˜π‘‰) ∧ (β„Žβ€˜π‘₯) ∈ 𝑉 ∧ (π‘“β€˜π‘₯) ∈ 𝑉) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ ℝ*)
9479, 92, 82, 93syl3anc 1371 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ ℝ*)
95 simp3l 1201 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (β„Žπ·π‘“) ∈ ℝ)
9695adantr 481 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (β„Žπ·π‘“) ∈ ℝ)
97 xmetge0 23619 . . . . . . . . . . . 12 ((𝐸 ∈ (∞Metβ€˜π‘‰) ∧ (β„Žβ€˜π‘₯) ∈ 𝑉 ∧ (π‘“β€˜π‘₯) ∈ 𝑉) β†’ 0 ≀ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)))
9879, 92, 82, 97syl3anc 1371 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ 0 ≀ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)))
9994fmpttd 7057 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))):πΌβŸΆβ„*)
10099frnd 6671 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βŠ† ℝ*)
10137a1i 11 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 0 ∈ ℝ*)
102101snssd 4767 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ {0} βŠ† ℝ*)
103100, 102unssd 4144 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ*)
104 ssun1 4130 . . . . . . . . . . . . . 14 ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βŠ† (ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βˆͺ {0})
105 ovex 7382 . . . . . . . . . . . . . . . . 17 ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ V
106105elabrex 7184 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝐼 β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))})
107106adantl 482 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))})
108 eqid 2737 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)))
109108rnmpt 5906 . . . . . . . . . . . . . . 15 ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))}
110107, 109eleqtrrdi 2849 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))))
111104, 110sselid 3940 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βˆͺ {0}))
112 supxrub 13171 . . . . . . . . . . . . 13 (((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ* ∧ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βˆͺ {0})) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ≀ sup((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
113103, 111, 112syl2an2r 683 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ≀ sup((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
114 simp21 1206 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 𝑓 ∈ 𝐡)
1154, 1, 87, 88, 89, 90, 114, 5, 6, 7prdsdsval3 17301 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (β„Žπ·π‘“) = sup((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
116115adantr 481 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (β„Žπ·π‘“) = sup((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
117113, 116breqtrrd 5131 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ≀ (β„Žπ·π‘“))
118 xrrege0 13021 . . . . . . . . . . 11 (((((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ ℝ* ∧ (β„Žπ·π‘“) ∈ ℝ) ∧ (0 ≀ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∧ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ≀ (β„Žπ·π‘“))) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ ℝ)
11994, 96, 98, 117, 118syl22anc 837 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ ℝ)
120 xmetcl 23606 . . . . . . . . . . . 12 ((𝐸 ∈ (∞Metβ€˜π‘‰) ∧ (β„Žβ€˜π‘₯) ∈ 𝑉 ∧ (π‘”β€˜π‘₯) ∈ 𝑉) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ*)
12179, 92, 85, 120syl3anc 1371 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ*)
122 simp3r 1202 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (β„Žπ·π‘”) ∈ ℝ)
123122adantr 481 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (β„Žπ·π‘”) ∈ ℝ)
124 xmetge0 23619 . . . . . . . . . . . 12 ((𝐸 ∈ (∞Metβ€˜π‘‰) ∧ (β„Žβ€˜π‘₯) ∈ 𝑉 ∧ (π‘”β€˜π‘₯) ∈ 𝑉) β†’ 0 ≀ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
12579, 92, 85, 124syl3anc 1371 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ 0 ≀ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
126121fmpttd 7057 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))):πΌβŸΆβ„*)
127126frnd 6671 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βŠ† ℝ*)
128127, 102unssd 4144 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ*)
129 ssun1 4130 . . . . . . . . . . . . . 14 ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βŠ† (ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})
130 ovex 7382 . . . . . . . . . . . . . . . . 17 ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ V
131130elabrex 7184 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝐼 β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))})
132131adantl 482 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))})
133 eqid 2737 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
134133rnmpt 5906 . . . . . . . . . . . . . . 15 ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))}
135132, 134eleqtrrdi 2849 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))))
136129, 135sselid 3940 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}))
137 supxrub 13171 . . . . . . . . . . . . 13 (((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ* ∧ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ sup((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
138128, 136, 137syl2an2r 683 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ sup((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
139 simp22 1207 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 𝑔 ∈ 𝐡)
1404, 1, 87, 88, 89, 90, 139, 5, 6, 7prdsdsval3 17301 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (β„Žπ·π‘”) = sup((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
141140adantr 481 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (β„Žπ·π‘”) = sup((ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
142138, 141breqtrrd 5131 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ (β„Žπ·π‘”))
143 xrrege0 13021 . . . . . . . . . . 11 (((((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ* ∧ (β„Žπ·π‘”) ∈ ℝ) ∧ (0 ≀ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∧ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ (β„Žπ·π‘”))) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ)
144121, 123, 125, 142, 143syl22anc 837 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ)
145119, 144readdcld 11117 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) + ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) ∈ ℝ)
14679, 82, 85, 57syl3anc 1371 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ 0 ≀ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
147 xmettri2 23615 . . . . . . . . . . 11 ((𝐸 ∈ (∞Metβ€˜π‘‰) ∧ ((β„Žβ€˜π‘₯) ∈ 𝑉 ∧ (π‘“β€˜π‘₯) ∈ 𝑉 ∧ (π‘”β€˜π‘₯) ∈ 𝑉)) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ (((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) +𝑒 ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))))
14879, 92, 82, 85, 147syl13anc 1372 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ (((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) +𝑒 ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))))
149119, 144rexaddd 13081 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) +𝑒 ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) = (((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) + ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))))
150148, 149breqtrd 5129 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ (((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) + ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))))
151 xrrege0 13021 . . . . . . . . 9 (((((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ* ∧ (((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) + ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) ∈ ℝ) ∧ (0 ≀ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∧ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ (((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) + ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))))) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ)
15286, 145, 146, 150, 151syl22anc 837 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ)
153 readdcl 11067 . . . . . . . . . 10 (((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ) β†’ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ∈ ℝ)
1541533ad2ant3 1135 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ∈ ℝ)
155154adantr 481 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ∈ ℝ)
156119, 144, 96, 123, 117, 142le2addd 11707 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ (((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) + ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
157152, 145, 155, 150, 156letrd 11245 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) ∧ π‘₯ ∈ 𝐼) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
158157ralrimiva 3141 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
15986ralrimiva 3141 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ*)
160 breq1 5106 . . . . . . . 8 (𝑧 = ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) β†’ (𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ↔ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”))))
16152, 160ralrnmptw 7038 . . . . . . 7 (βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ* β†’ (βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ↔ βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”))))
162159, 161syl 17 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ↔ βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”))))
163158, 162mpbird 256 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
164123ad2ant1 1133 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 𝐷:(𝐡 Γ— 𝐡)⟢(0[,]+∞))
165164, 90, 114fovcdmd 7518 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (β„Žπ·π‘“) ∈ (0[,]+∞))
166 elxrge0 13302 . . . . . . . . 9 ((β„Žπ·π‘“) ∈ (0[,]+∞) ↔ ((β„Žπ·π‘“) ∈ ℝ* ∧ 0 ≀ (β„Žπ·π‘“)))
167166simprbi 497 . . . . . . . 8 ((β„Žπ·π‘“) ∈ (0[,]+∞) β†’ 0 ≀ (β„Žπ·π‘“))
168165, 167syl 17 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 0 ≀ (β„Žπ·π‘“))
169164, 90, 139fovcdmd 7518 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (β„Žπ·π‘”) ∈ (0[,]+∞))
170 elxrge0 13302 . . . . . . . . 9 ((β„Žπ·π‘”) ∈ (0[,]+∞) ↔ ((β„Žπ·π‘”) ∈ ℝ* ∧ 0 ≀ (β„Žπ·π‘”)))
171170simprbi 497 . . . . . . . 8 ((β„Žπ·π‘”) ∈ (0[,]+∞) β†’ 0 ≀ (β„Žπ·π‘”))
172169, 171syl 17 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 0 ≀ (β„Žπ·π‘”))
17395, 122, 168, 172addge0d 11664 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 0 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
174 breq1 5106 . . . . . . 7 (𝑧 = 0 β†’ (𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ↔ 0 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”))))
17544, 174ralsn 4640 . . . . . 6 (βˆ€π‘§ ∈ {0}𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ↔ 0 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
176173, 175sylibr 233 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ βˆ€π‘§ ∈ {0}𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
177 ralunb 4149 . . . . 5 (βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ↔ (βˆ€π‘§ ∈ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ∧ βˆ€π‘§ ∈ {0}𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”))))
178163, 176, 177sylanbrc 583 . . . 4 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
179403adantr3 1171 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡)) β†’ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ*)
1801793adant3 1132 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ*)
181154rexrd 11138 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ∈ ℝ*)
182 supxrleub 13173 . . . . 5 (((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ* ∧ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ∈ ℝ*) β†’ (sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ↔ βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”))))
183180, 181, 182syl2anc 584 . . . 4 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ↔ βˆ€π‘§ ∈ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0})𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”))))
184178, 183mpbird 256 . . 3 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
18578, 184eqbrtrd 5125 . 2 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (𝑓𝐷𝑔) ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)))
1863, 15, 19, 76, 185isxmet2d 23602 1 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2714  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3443   βˆͺ cun 3906   βŠ† wss 3908  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   Γ— cxp 5628  ran crn 5631   β†Ύ cres 5632   Fn wfn 6486  βŸΆwf 6487  β€˜cfv 6491  (class class class)co 7349  supcsup 9309  β„cr 10983  0cc0 10984   + caddc 10987  +∞cpnf 11119  β„*cxr 11121   < clt 11122   ≀ cle 11123   +𝑒 cxad 12959  [,]cicc 13195  Basecbs 17017  distcds 17076  Xscprds 17261  βˆžMetcxmet 20704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7662  ax-cnex 11040  ax-resscn 11041  ax-1cn 11042  ax-icn 11043  ax-addcl 11044  ax-addrcl 11045  ax-mulcl 11046  ax-mulrcl 11047  ax-mulcom 11048  ax-addass 11049  ax-mulass 11050  ax-distr 11051  ax-i2m1 11052  ax-1ne0 11053  ax-1rid 11054  ax-rnegex 11055  ax-rrecex 11056  ax-cnre 11057  ax-pre-lttri 11058  ax-pre-lttrn 11059  ax-pre-ltadd 11060  ax-pre-mulgt0 11061  ax-pre-sup 11062
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5528  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5585  df-we 5587  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-rn 5641  df-res 5642  df-ima 5643  df-pred 6249  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7305  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7793  df-1st 7911  df-2nd 7912  df-frecs 8179  df-wrecs 8210  df-recs 8284  df-rdg 8323  df-1o 8379  df-er 8581  df-map 8700  df-ixp 8769  df-en 8817  df-dom 8818  df-sdom 8819  df-fin 8820  df-sup 9311  df-pnf 11124  df-mnf 11125  df-xr 11126  df-ltxr 11127  df-le 11128  df-sub 11320  df-neg 11321  df-div 11746  df-nn 12087  df-2 12149  df-3 12150  df-4 12151  df-5 12152  df-6 12153  df-7 12154  df-8 12155  df-9 12156  df-n0 12347  df-z 12433  df-dec 12551  df-uz 12696  df-rp 12844  df-xneg 12961  df-xadd 12962  df-xmul 12963  df-icc 13199  df-fz 13353  df-struct 16953  df-slot 16988  df-ndx 17000  df-base 17018  df-plusg 17080  df-mulr 17081  df-sca 17083  df-vsca 17084  df-ip 17085  df-tset 17086  df-ple 17087  df-ds 17089  df-hom 17091  df-cco 17092  df-prds 17263  df-xmet 20712
This theorem is referenced by:  prdsxmet  23644
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