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Theorem dvnres 25448
Description: Multiple derivative version of dvres3a 25431. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
dvnres (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚) ∧ 𝑁 ∈ β„•0) ∧ dom ((β„‚ D𝑛 𝐹)β€˜π‘) = dom 𝐹) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆))

Proof of Theorem dvnres
Dummy variables 𝑛 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6892 . . . . . . . . 9 (π‘₯ = 0 β†’ ((β„‚ D𝑛 𝐹)β€˜π‘₯) = ((β„‚ D𝑛 𝐹)β€˜0))
21dmeqd 5906 . . . . . . . 8 (π‘₯ = 0 β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom ((β„‚ D𝑛 𝐹)β€˜0))
32eqeq1d 2735 . . . . . . 7 (π‘₯ = 0 β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 ↔ dom ((β„‚ D𝑛 𝐹)β€˜0) = dom 𝐹))
4 fveq2 6892 . . . . . . . 8 (π‘₯ = 0 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0))
51reseq1d 5981 . . . . . . . 8 (π‘₯ = 0 β†’ (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) = (((β„‚ D𝑛 𝐹)β€˜0) β†Ύ 𝑆))
64, 5eqeq12d 2749 . . . . . . 7 (π‘₯ = 0 β†’ (((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) ↔ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0) = (((β„‚ D𝑛 𝐹)β€˜0) β†Ύ 𝑆)))
73, 6imbi12d 345 . . . . . 6 (π‘₯ = 0 β†’ ((dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆)) ↔ (dom ((β„‚ D𝑛 𝐹)β€˜0) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0) = (((β„‚ D𝑛 𝐹)β€˜0) β†Ύ 𝑆))))
87imbi2d 341 . . . . 5 (π‘₯ = 0 β†’ (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆))) ↔ ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜0) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0) = (((β„‚ D𝑛 𝐹)β€˜0) β†Ύ 𝑆)))))
9 fveq2 6892 . . . . . . . . 9 (π‘₯ = 𝑛 β†’ ((β„‚ D𝑛 𝐹)β€˜π‘₯) = ((β„‚ D𝑛 𝐹)β€˜π‘›))
109dmeqd 5906 . . . . . . . 8 (π‘₯ = 𝑛 β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom ((β„‚ D𝑛 𝐹)β€˜π‘›))
1110eqeq1d 2735 . . . . . . 7 (π‘₯ = 𝑛 β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 ↔ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) = dom 𝐹))
12 fveq2 6892 . . . . . . . 8 (π‘₯ = 𝑛 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›))
139reseq1d 5981 . . . . . . . 8 (π‘₯ = 𝑛 β†’ (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆))
1412, 13eqeq12d 2749 . . . . . . 7 (π‘₯ = 𝑛 β†’ (((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) ↔ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆)))
1511, 14imbi12d 345 . . . . . 6 (π‘₯ = 𝑛 β†’ ((dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆)) ↔ (dom ((β„‚ D𝑛 𝐹)β€˜π‘›) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆))))
1615imbi2d 341 . . . . 5 (π‘₯ = 𝑛 β†’ (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆))) ↔ ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘›) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆)))))
17 fveq2 6892 . . . . . . . . 9 (π‘₯ = (𝑛 + 1) β†’ ((β„‚ D𝑛 𝐹)β€˜π‘₯) = ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)))
1817dmeqd 5906 . . . . . . . 8 (π‘₯ = (𝑛 + 1) β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)))
1918eqeq1d 2735 . . . . . . 7 (π‘₯ = (𝑛 + 1) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 ↔ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹))
20 fveq2 6892 . . . . . . . 8 (π‘₯ = (𝑛 + 1) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)))
2117reseq1d 5981 . . . . . . . 8 (π‘₯ = (𝑛 + 1) β†’ (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) = (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆))
2220, 21eqeq12d 2749 . . . . . . 7 (π‘₯ = (𝑛 + 1) β†’ (((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) ↔ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)) = (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆)))
2319, 22imbi12d 345 . . . . . 6 (π‘₯ = (𝑛 + 1) β†’ ((dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆)) ↔ (dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)) = (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆))))
2423imbi2d 341 . . . . 5 (π‘₯ = (𝑛 + 1) β†’ (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆))) ↔ ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)) = (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆)))))
25 fveq2 6892 . . . . . . . . 9 (π‘₯ = 𝑁 β†’ ((β„‚ D𝑛 𝐹)β€˜π‘₯) = ((β„‚ D𝑛 𝐹)β€˜π‘))
2625dmeqd 5906 . . . . . . . 8 (π‘₯ = 𝑁 β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom ((β„‚ D𝑛 𝐹)β€˜π‘))
2726eqeq1d 2735 . . . . . . 7 (π‘₯ = 𝑁 β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 ↔ dom ((β„‚ D𝑛 𝐹)β€˜π‘) = dom 𝐹))
28 fveq2 6892 . . . . . . . 8 (π‘₯ = 𝑁 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘))
2925reseq1d 5981 . . . . . . . 8 (π‘₯ = 𝑁 β†’ (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆))
3028, 29eqeq12d 2749 . . . . . . 7 (π‘₯ = 𝑁 β†’ (((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) ↔ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆)))
3127, 30imbi12d 345 . . . . . 6 (π‘₯ = 𝑁 β†’ ((dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆)) ↔ (dom ((β„‚ D𝑛 𝐹)β€˜π‘) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆))))
3231imbi2d 341 . . . . 5 (π‘₯ = 𝑁 β†’ (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆))) ↔ ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆)))))
33 recnprss 25421 . . . . . . . . 9 (𝑆 ∈ {ℝ, β„‚} β†’ 𝑆 βŠ† β„‚)
3433adantr 482 . . . . . . . 8 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ 𝑆 βŠ† β„‚)
35 pmresg 8864 . . . . . . . 8 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (𝐹 β†Ύ 𝑆) ∈ (β„‚ ↑pm 𝑆))
36 dvn0 25441 . . . . . . . 8 ((𝑆 βŠ† β„‚ ∧ (𝐹 β†Ύ 𝑆) ∈ (β„‚ ↑pm 𝑆)) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0) = (𝐹 β†Ύ 𝑆))
3734, 35, 36syl2anc 585 . . . . . . 7 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0) = (𝐹 β†Ύ 𝑆))
38 ssidd 4006 . . . . . . . . 9 (𝑆 ∈ {ℝ, β„‚} β†’ β„‚ βŠ† β„‚)
39 dvn0 25441 . . . . . . . . 9 ((β„‚ βŠ† β„‚ ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ ((β„‚ D𝑛 𝐹)β€˜0) = 𝐹)
4038, 39sylan 581 . . . . . . . 8 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ ((β„‚ D𝑛 𝐹)β€˜0) = 𝐹)
4140reseq1d 5981 . . . . . . 7 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (((β„‚ D𝑛 𝐹)β€˜0) β†Ύ 𝑆) = (𝐹 β†Ύ 𝑆))
4237, 41eqtr4d 2776 . . . . . 6 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0) = (((β„‚ D𝑛 𝐹)β€˜0) β†Ύ 𝑆))
4342a1d 25 . . . . 5 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜0) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0) = (((β„‚ D𝑛 𝐹)β€˜0) β†Ύ 𝑆)))
44 cnelprrecn 11203 . . . . . . . . . 10 β„‚ ∈ {ℝ, β„‚}
45 simplr 768 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ 𝐹 ∈ (β„‚ ↑pm β„‚))
46 simprl 770 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ 𝑛 ∈ β„•0)
47 dvnbss 25445 . . . . . . . . . 10 ((β„‚ ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚) ∧ 𝑛 ∈ β„•0) β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) βŠ† dom 𝐹)
4844, 45, 46, 47mp3an2i 1467 . . . . . . . . 9 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) βŠ† dom 𝐹)
49 simprr 772 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)
50 ssidd 4006 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ β„‚ βŠ† β„‚)
51 dvnp1 25442 . . . . . . . . . . . . 13 ((β„‚ βŠ† β„‚ ∧ 𝐹 ∈ (β„‚ ↑pm β„‚) ∧ 𝑛 ∈ β„•0) β†’ ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = (β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)))
5250, 45, 46, 51syl3anc 1372 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = (β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)))
5352dmeqd 5906 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom (β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)))
5449, 53eqtr3d 2775 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom 𝐹 = dom (β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)))
55 dvnf 25444 . . . . . . . . . . . 12 ((β„‚ ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚) ∧ 𝑛 ∈ β„•0) β†’ ((β„‚ D𝑛 𝐹)β€˜π‘›):dom ((β„‚ D𝑛 𝐹)β€˜π‘›)βŸΆβ„‚)
5644, 45, 46, 55mp3an2i 1467 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ ((β„‚ D𝑛 𝐹)β€˜π‘›):dom ((β„‚ D𝑛 𝐹)β€˜π‘›)βŸΆβ„‚)
57 cnex 11191 . . . . . . . . . . . . . . 15 β„‚ ∈ V
5857, 57elpm2 8868 . . . . . . . . . . . . . 14 (𝐹 ∈ (β„‚ ↑pm β„‚) ↔ (𝐹:dom πΉβŸΆβ„‚ ∧ dom 𝐹 βŠ† β„‚))
5958simprbi 498 . . . . . . . . . . . . 13 (𝐹 ∈ (β„‚ ↑pm β„‚) β†’ dom 𝐹 βŠ† β„‚)
6045, 59syl 17 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom 𝐹 βŠ† β„‚)
6148, 60sstrd 3993 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) βŠ† β„‚)
6250, 56, 61dvbss 25418 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom (β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)) βŠ† dom ((β„‚ D𝑛 𝐹)β€˜π‘›))
6354, 62eqsstrd 4021 . . . . . . . . 9 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom 𝐹 βŠ† dom ((β„‚ D𝑛 𝐹)β€˜π‘›))
6448, 63eqssd 4000 . . . . . . . 8 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) = dom 𝐹)
6564expr 458 . . . . . . 7 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ 𝑛 ∈ β„•0) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹 β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) = dom 𝐹))
6665imim1d 82 . . . . . 6 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ 𝑛 ∈ β„•0) β†’ ((dom ((β„‚ D𝑛 𝐹)β€˜π‘›) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆))))
67 oveq2 7417 . . . . . . 7 (((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆) β†’ (𝑆 D ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›)) = (𝑆 D (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆)))
6834adantr 482 . . . . . . . . 9 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ 𝑆 βŠ† β„‚)
6935adantr 482 . . . . . . . . 9 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ (𝐹 β†Ύ 𝑆) ∈ (β„‚ ↑pm 𝑆))
70 dvnp1 25442 . . . . . . . . 9 ((𝑆 βŠ† β„‚ ∧ (𝐹 β†Ύ 𝑆) ∈ (β„‚ ↑pm 𝑆) ∧ 𝑛 ∈ β„•0) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›)))
7168, 69, 46, 70syl3anc 1372 . . . . . . . 8 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›)))
7252reseq1d 5981 . . . . . . . . 9 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆) = ((β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)) β†Ύ 𝑆))
73 simpll 766 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ 𝑆 ∈ {ℝ, β„‚})
74 eqid 2733 . . . . . . . . . . . . . 14 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
7574cnfldtop 24300 . . . . . . . . . . . . 13 (TopOpenβ€˜β„‚fld) ∈ Top
76 unicntop 24302 . . . . . . . . . . . . . 14 β„‚ = βˆͺ (TopOpenβ€˜β„‚fld)
7776ntrss2 22561 . . . . . . . . . . . . 13 (((TopOpenβ€˜β„‚fld) ∈ Top ∧ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) βŠ† β„‚) β†’ ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜dom ((β„‚ D𝑛 𝐹)β€˜π‘›)) βŠ† dom ((β„‚ D𝑛 𝐹)β€˜π‘›))
7875, 61, 77sylancr 588 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜dom ((β„‚ D𝑛 𝐹)β€˜π‘›)) βŠ† dom ((β„‚ D𝑛 𝐹)β€˜π‘›))
7974cnfldtopon 24299 . . . . . . . . . . . . . . . 16 (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚)
8079toponrestid 22423 . . . . . . . . . . . . . . 15 (TopOpenβ€˜β„‚fld) = ((TopOpenβ€˜β„‚fld) β†Ύt β„‚)
8150, 56, 61, 80, 74dvbssntr 25417 . . . . . . . . . . . . . 14 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom (β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)) βŠ† ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜dom ((β„‚ D𝑛 𝐹)β€˜π‘›)))
8254, 81eqsstrd 4021 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom 𝐹 βŠ† ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜dom ((β„‚ D𝑛 𝐹)β€˜π‘›)))
8348, 82sstrd 3993 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) βŠ† ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜dom ((β„‚ D𝑛 𝐹)β€˜π‘›)))
8478, 83eqssd 4000 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜dom ((β„‚ D𝑛 𝐹)β€˜π‘›)) = dom ((β„‚ D𝑛 𝐹)β€˜π‘›))
8576isopn3 22570 . . . . . . . . . . . 12 (((TopOpenβ€˜β„‚fld) ∈ Top ∧ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) βŠ† β„‚) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘›) ∈ (TopOpenβ€˜β„‚fld) ↔ ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜dom ((β„‚ D𝑛 𝐹)β€˜π‘›)) = dom ((β„‚ D𝑛 𝐹)β€˜π‘›)))
8675, 61, 85sylancr 588 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘›) ∈ (TopOpenβ€˜β„‚fld) ↔ ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜dom ((β„‚ D𝑛 𝐹)β€˜π‘›)) = dom ((β„‚ D𝑛 𝐹)β€˜π‘›)))
8784, 86mpbird 257 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) ∈ (TopOpenβ€˜β„‚fld))
8864, 54eqtr2d 2774 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ dom (β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)) = dom ((β„‚ D𝑛 𝐹)β€˜π‘›))
8974dvres3a 25431 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, β„‚} ∧ ((β„‚ D𝑛 𝐹)β€˜π‘›):dom ((β„‚ D𝑛 𝐹)β€˜π‘›)βŸΆβ„‚) ∧ (dom ((β„‚ D𝑛 𝐹)β€˜π‘›) ∈ (TopOpenβ€˜β„‚fld) ∧ dom (β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)) = dom ((β„‚ D𝑛 𝐹)β€˜π‘›))) β†’ (𝑆 D (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆)) = ((β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)) β†Ύ 𝑆))
9073, 56, 87, 88, 89syl22anc 838 . . . . . . . . 9 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ (𝑆 D (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆)) = ((β„‚ D ((β„‚ D𝑛 𝐹)β€˜π‘›)) β†Ύ 𝑆))
9172, 90eqtr4d 2776 . . . . . . . 8 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆) = (𝑆 D (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆)))
9271, 91eqeq12d 2749 . . . . . . 7 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ (((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)) = (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆) ↔ (𝑆 D ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›)) = (𝑆 D (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆))))
9367, 92imbitrrid 245 . . . . . 6 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) ∧ (𝑛 ∈ β„•0 ∧ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹)) β†’ (((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)) = (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆)))
9466, 93animpimp2impd 845 . . . . 5 (𝑛 ∈ β„•0 β†’ (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘›) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆))) β†’ ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)) = (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆)))))
958, 16, 24, 32, 43, 94nn0ind 12657 . . . 4 (𝑁 ∈ β„•0 β†’ ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆))))
9695com12 32 . . 3 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (𝑁 ∈ β„•0 β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆))))
97963impia 1118 . 2 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚) ∧ 𝑁 ∈ β„•0) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘) = dom 𝐹 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆)))
9897imp 408 1 (((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚) ∧ 𝑁 ∈ β„•0) ∧ dom ((β„‚ D𝑛 𝐹)β€˜π‘) = dom 𝐹) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  {cpr 4631  dom cdm 5677   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑pm cpm 8821  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113  β„•0cn0 12472  TopOpenctopn 17367  β„‚fldccnfld 20944  Topctop 22395  intcnt 22521   D cdv 25380   D𝑛 cdvn 25381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-icc 13331  df-fz 13485  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-mulr 17211  df-starv 17212  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-rest 17368  df-topn 17369  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-fbas 20941  df-fg 20942  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-lp 22640  df-perf 22641  df-cnp 22732  df-haus 22819  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444  df-xms 23826  df-ms 23827  df-limc 25383  df-dv 25384  df-dvn 25385
This theorem is referenced by:  cpnres  25454
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