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Theorem prdsxmetlem 24196
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 6895 . . 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 24195 . . 3 (πœ‘ β†’ 𝐷:(𝐡 Γ— 𝐡)⟢(0[,]+∞))
13 iccssxr 13404 . . 3 (0[,]+∞) βŠ† ℝ*
14 fss 6724 . . 3 ((𝐷:(𝐡 Γ— 𝐡)⟢(0[,]+∞) ∧ (0[,]+∞) βŠ† ℝ*) β†’ 𝐷:(𝐡 Γ— 𝐡)βŸΆβ„*)
1512, 13, 14sylancl 585 . 2 (πœ‘ β†’ 𝐷:(𝐡 Γ— 𝐡)βŸΆβ„*)
1612fovcdmda 7571 . . 3 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑓𝐷𝑔) ∈ (0[,]+∞))
17 elxrge0 13431 . . . 4 ((𝑓𝐷𝑔) ∈ (0[,]+∞) ↔ ((𝑓𝐷𝑔) ∈ ℝ* ∧ 0 ≀ (𝑓𝐷𝑔)))
1817simprbi 496 . . 3 ((𝑓𝐷𝑔) ∈ (0[,]+∞) β†’ 0 ≀ (𝑓𝐷𝑔))
1916, 18syl 17 . 2 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 0 ≀ (𝑓𝐷𝑔))
208adantr 480 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑆 ∈ π‘Š)
219adantr 480 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝐼 ∈ 𝑋)
2210ralrimiva 3138 . . . . . 6 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐼 𝑅 ∈ 𝑍)
2322adantr 480 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 𝑅 ∈ 𝑍)
24 simprl 768 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑓 ∈ 𝐡)
25 simprr 770 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑔 ∈ 𝐡)
264, 1, 20, 21, 23, 24, 25, 5, 6, 7prdsdsval3 17430 . . . 4 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑓𝐷𝑔) = sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))
2726breq1d 5148 . . 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 712 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ 𝐸 ∈ (∞Metβ€˜π‘‰))
294, 1, 20, 21, 23, 5, 24prdsbascl 17428 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) ∈ 𝑉)
3029r19.21bi 3240 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ (π‘“β€˜π‘₯) ∈ 𝑉)
314, 1, 20, 21, 23, 5, 25prdsbascl 17428 . . . . . . . . 9 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘”β€˜π‘₯) ∈ 𝑉)
3231r19.21bi 3240 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ (π‘”β€˜π‘₯) ∈ 𝑉)
33 xmetcl 24159 . . . . . . . 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 1368 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ ℝ*)
3534fmpttd 7106 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))):πΌβŸΆβ„*)
3635frnd 6715 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βŠ† ℝ*)
37 0xr 11258 . . . . . . 7 0 ∈ ℝ*
3837a1i 11 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 0 ∈ ℝ*)
3938snssd 4804 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ {0} βŠ† ℝ*)
4036, 39unssd 4178 . . . 4 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) βˆͺ {0}) βŠ† ℝ*)
41 supxrleub 13302 . . . 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 585 . . 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 12310 . . . . . . 7 0 ≀ 0
44 c0ex 11205 . . . . . . . 8 0 ∈ V
45 breq1 5141 . . . . . . . 8 (𝑧 = 0 β†’ (𝑧 ≀ 0 ↔ 0 ≀ 0))
4644, 45ralsn 4677 . . . . . . 7 (βˆ€π‘§ ∈ {0}𝑧 ≀ 0 ↔ 0 ≀ 0)
4743, 46mpbir 230 . . . . . 6 βˆ€π‘§ ∈ {0}𝑧 ≀ 0
48 ralunb 4183 . . . . . 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 707 . . . . 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 7434 . . . . . . 7 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ V
5150rgenw 3057 . . . . . 6 βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ V
52 eqid 2724 . . . . . . 7 (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
53 breq1 5141 . . . . . . 7 (𝑧 = ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) β†’ (𝑧 ≀ 0 ↔ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0))
5452, 53ralrnmptw 7085 . . . . . 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 275 . . . 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 24172 . . . . . . . . 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 1368 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) ∧ π‘₯ ∈ 𝐼) β†’ 0 ≀ ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
5958biantrud 531 . . . . . . 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 13130 . . . . . . . 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 585 . . . . . . 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 24166 . . . . . . . 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 1368 . . . . . . 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 307 . . . . . 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 3167 . . . . 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 2724 . . . . . . . . . 10 (π‘₯ ∈ 𝐼 ↦ 𝑅) = (π‘₯ ∈ 𝐼 ↦ 𝑅)
6766fnmpt 6680 . . . . . . . . 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 480 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (π‘₯ ∈ 𝐼 ↦ 𝑅) Fn 𝐼)
704, 1, 20, 21, 69, 24prdsbasfn 17416 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑓 Fn 𝐼)
714, 1, 20, 21, 69, 25prdsbasfn 17416 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ 𝑔 Fn 𝐼)
72 eqfnfv 7022 . . . . . 6 ((𝑓 Fn 𝐼 ∧ 𝑔 Fn 𝐼) β†’ (𝑓 = 𝑔 ↔ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯)))
7370, 71, 72syl2anc 583 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑓 = 𝑔 ↔ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯)))
7465, 73bitr4d 282 . . . 4 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (βˆ€π‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ≀ 0 ↔ 𝑓 = 𝑔))
7556, 74bitrid 283 . . 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 305 . 2 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ ((𝑓𝐷𝑔) ≀ 0 ↔ 𝑓 = 𝑔))
77263adantr3 1168 . . . 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 1129 . . 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 1182 . . . . . . . . . 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 1168 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯) ∈ 𝑉)
81803adant3 1129 . . . . . . . . . . 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 3240 . . . . . . . . . 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 1168 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡)) β†’ βˆ€π‘₯ ∈ 𝐼 (π‘”β€˜π‘₯) ∈ 𝑉)
84833adant3 1129 . . . . . . . . . . 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 3240 . . . . . . . . . 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 1368 . . . . . . . . 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 1130 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 𝑆 ∈ π‘Š)
8893ad2ant1 1130 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ 𝐼 ∈ 𝑋)
89223ad2ant1 1130 . . . . . . . . . . . . . 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 1205 . . . . . . . . . . . . . 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 17428 . . . . . . . . . . . . 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 3240 . . . . . . . . . . . 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 24159 . . . . . . . . . . . 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 1368 . . . . . . . . . . 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 1198 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (β„Žπ·π‘“) ∈ ℝ)
9695adantr 480 . . . . . . . . . . 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 24172 . . . . . . . . . . . 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 1368 . . . . . . . . . . 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 7106 . . . . . . . . . . . . . . 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 6715 . . . . . . . . . . . . . 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 4804 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ {0} βŠ† ℝ*)
103100, 102unssd 4178 . . . . . . . . . . . . 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 4164 . . . . . . . . . . . . . 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 7434 . . . . . . . . . . . . . . . . 17 ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ V
106105elabrex 7233 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝐼 β†’ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)) ∈ {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))})
107106adantl 481 . . . . . . . . . . . . . . 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 2724 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯)))
109108rnmpt 5944 . . . . . . . . . . . . . . 15 ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))) = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘“β€˜π‘₯))}
110107, 109eleqtrrdi 2836 . . . . . . . . . . . . . 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 3972 . . . . . . . . . . . . 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 13300 . . . . . . . . . . . . 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 682 . . . . . . . . . . . 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 1203 . . . . . . . . . . . . . 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 17430 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 5166 . . . . . . . . . . 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 13150 . . . . . . . . . . 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 836 . . . . . . . . . 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 24159 . . . . . . . . . . . 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 1368 . . . . . . . . . . 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 1199 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡 ∧ β„Ž ∈ 𝐡) ∧ ((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ)) β†’ (β„Žπ·π‘”) ∈ ℝ)
123122adantr 480 . . . . . . . . . . 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 24172 . . . . . . . . . . . 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 1368 . . . . . . . . . . 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 7106 . . . . . . . . . . . . . . 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 6715 . . . . . . . . . . . . . 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 4178 . . . . . . . . . . . . 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 4164 . . . . . . . . . . . . . 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 7434 . . . . . . . . . . . . . . . . 17 ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ V
131130elabrex 7233 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝐼 β†’ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)) ∈ {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))})
132131adantl 481 . . . . . . . . . . . . . . 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 2724 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯)))
134133rnmpt 5944 . . . . . . . . . . . . . . 15 ran (π‘₯ ∈ 𝐼 ↦ ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))) = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐼 𝑧 = ((β„Žβ€˜π‘₯)𝐸(π‘”β€˜π‘₯))}
135132, 134eleqtrrdi 2836 . . . . . . . . . . . . . 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 3972 . . . . . . . . . . . . 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 13300 . . . . . . . . . . . . 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 682 . . . . . . . . . . . 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 1204 . . . . . . . . . . . . . 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 17430 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 5166 . . . . . . . . . . 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 13150 . . . . . . . . . . 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 836 . . . . . . . . . 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 11240 . . . . . . . . 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 1368 . . . . . . . . 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 24168 . . . . . . . . . . 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 1369 . . . . . . . . . 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 13210 . . . . . . . . . 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 5164 . . . . . . . . 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 13150 . . . . . . . . 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 836 . . . . . . . 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 11189 . . . . . . . . . 10 (((β„Žπ·π‘“) ∈ ℝ ∧ (β„Žπ·π‘”) ∈ ℝ) β†’ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ∈ ℝ)
1541533ad2ant3 1132 . . . . . . . . 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 480 . . . . . . . 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 11830 . . . . . . . 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 11368 . . . . . . 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 3138 . . . . . 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 3138 . . . . . . 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 5141 . . . . . . . 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 7085 . . . . . . 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 257 . . . . 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 1130 . . . . . . . . 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 7572 . . . . . . . 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 13431 . . . . . . . . 9 ((β„Žπ·π‘“) ∈ (0[,]+∞) ↔ ((β„Žπ·π‘“) ∈ ℝ* ∧ 0 ≀ (β„Žπ·π‘“)))
167166simprbi 496 . . . . . . . 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 7572 . . . . . . . 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 13431 . . . . . . . . 9 ((β„Žπ·π‘”) ∈ (0[,]+∞) ↔ ((β„Žπ·π‘”) ∈ ℝ* ∧ 0 ≀ (β„Žπ·π‘”)))
171170simprbi 496 . . . . . . . 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 11787 . . . . . 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 5141 . . . . . . 7 (𝑧 = 0 β†’ (𝑧 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”)) ↔ 0 ≀ ((β„Žπ·π‘“) + (β„Žπ·π‘”))))
17544, 174ralsn 4677 . . . . . 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 4183 . . . . 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 582 . . . 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 1168 . . . . . 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 1129 . . . . 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 11261 . . . . 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 13302 . . . . 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 583 . . . 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 257 . . 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 5160 . 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 24155 1 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2701  βˆ€wral 3053  βˆƒwrex 3062  Vcvv 3466   βˆͺ cun 3938   βŠ† wss 3940  {csn 4620   class class class wbr 5138   ↦ cmpt 5221   Γ— cxp 5664  ran crn 5667   β†Ύ cres 5668   Fn wfn 6528  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401  supcsup 9431  β„cr 11105  0cc0 11106   + caddc 11109  +∞cpnf 11242  β„*cxr 11244   < clt 11245   ≀ cle 11246   +𝑒 cxad 13087  [,]cicc 13324  Basecbs 17143  distcds 17205  Xscprds 17390  βˆžMetcxmet 21213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-icc 13328  df-fz 13482  df-struct 17079  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-hom 17220  df-cco 17221  df-prds 17392  df-xmet 21221
This theorem is referenced by:  prdsxmet  24197
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