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Theorem dvnres 25672
Description: Multiple derivative version of dvres3a 25655. (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 6891 . . . . . . . . 9 (π‘₯ = 0 β†’ ((β„‚ D𝑛 𝐹)β€˜π‘₯) = ((β„‚ D𝑛 𝐹)β€˜0))
21dmeqd 5905 . . . . . . . 8 (π‘₯ = 0 β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom ((β„‚ D𝑛 𝐹)β€˜0))
32eqeq1d 2734 . . . . . . 7 (π‘₯ = 0 β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 ↔ dom ((β„‚ D𝑛 𝐹)β€˜0) = dom 𝐹))
4 fveq2 6891 . . . . . . . 8 (π‘₯ = 0 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0))
51reseq1d 5980 . . . . . . . 8 (π‘₯ = 0 β†’ (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) = (((β„‚ D𝑛 𝐹)β€˜0) β†Ύ 𝑆))
64, 5eqeq12d 2748 . . . . . . 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 344 . . . . . 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 340 . . . . 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 6891 . . . . . . . . 9 (π‘₯ = 𝑛 β†’ ((β„‚ D𝑛 𝐹)β€˜π‘₯) = ((β„‚ D𝑛 𝐹)β€˜π‘›))
109dmeqd 5905 . . . . . . . 8 (π‘₯ = 𝑛 β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom ((β„‚ D𝑛 𝐹)β€˜π‘›))
1110eqeq1d 2734 . . . . . . 7 (π‘₯ = 𝑛 β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 ↔ dom ((β„‚ D𝑛 𝐹)β€˜π‘›) = dom 𝐹))
12 fveq2 6891 . . . . . . . 8 (π‘₯ = 𝑛 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘›))
139reseq1d 5980 . . . . . . . 8 (π‘₯ = 𝑛 β†’ (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) = (((β„‚ D𝑛 𝐹)β€˜π‘›) β†Ύ 𝑆))
1412, 13eqeq12d 2748 . . . . . . 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 344 . . . . . 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 340 . . . . 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 6891 . . . . . . . . 9 (π‘₯ = (𝑛 + 1) β†’ ((β„‚ D𝑛 𝐹)β€˜π‘₯) = ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)))
1817dmeqd 5905 . . . . . . . 8 (π‘₯ = (𝑛 + 1) β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)))
1918eqeq1d 2734 . . . . . . 7 (π‘₯ = (𝑛 + 1) β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 ↔ dom ((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) = dom 𝐹))
20 fveq2 6891 . . . . . . . 8 (π‘₯ = (𝑛 + 1) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜(𝑛 + 1)))
2117reseq1d 5980 . . . . . . . 8 (π‘₯ = (𝑛 + 1) β†’ (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) = (((β„‚ D𝑛 𝐹)β€˜(𝑛 + 1)) β†Ύ 𝑆))
2220, 21eqeq12d 2748 . . . . . . 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 344 . . . . . 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 340 . . . . 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 6891 . . . . . . . . 9 (π‘₯ = 𝑁 β†’ ((β„‚ D𝑛 𝐹)β€˜π‘₯) = ((β„‚ D𝑛 𝐹)β€˜π‘))
2625dmeqd 5905 . . . . . . . 8 (π‘₯ = 𝑁 β†’ dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom ((β„‚ D𝑛 𝐹)β€˜π‘))
2726eqeq1d 2734 . . . . . . 7 (π‘₯ = 𝑁 β†’ (dom ((β„‚ D𝑛 𝐹)β€˜π‘₯) = dom 𝐹 ↔ dom ((β„‚ D𝑛 𝐹)β€˜π‘) = dom 𝐹))
28 fveq2 6891 . . . . . . . 8 (π‘₯ = 𝑁 β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘₯) = ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜π‘))
2925reseq1d 5980 . . . . . . . 8 (π‘₯ = 𝑁 β†’ (((β„‚ D𝑛 𝐹)β€˜π‘₯) β†Ύ 𝑆) = (((β„‚ D𝑛 𝐹)β€˜π‘) β†Ύ 𝑆))
3028, 29eqeq12d 2748 . . . . . . 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 344 . . . . . 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 340 . . . . 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 25645 . . . . . . . . 9 (𝑆 ∈ {ℝ, β„‚} β†’ 𝑆 βŠ† β„‚)
3433adantr 481 . . . . . . . 8 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ 𝑆 βŠ† β„‚)
35 pmresg 8866 . . . . . . . 8 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (𝐹 β†Ύ 𝑆) ∈ (β„‚ ↑pm 𝑆))
36 dvn0 25665 . . . . . . . 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 584 . . . . . . 7 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ ((𝑆 D𝑛 (𝐹 β†Ύ 𝑆))β€˜0) = (𝐹 β†Ύ 𝑆))
38 ssidd 4005 . . . . . . . . 9 (𝑆 ∈ {ℝ, β„‚} β†’ β„‚ βŠ† β„‚)
39 dvn0 25665 . . . . . . . . 9 ((β„‚ βŠ† β„‚ ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ ((β„‚ D𝑛 𝐹)β€˜0) = 𝐹)
4038, 39sylan 580 . . . . . . . 8 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ ((β„‚ D𝑛 𝐹)β€˜0) = 𝐹)
4140reseq1d 5980 . . . . . . 7 ((𝑆 ∈ {ℝ, β„‚} ∧ 𝐹 ∈ (β„‚ ↑pm β„‚)) β†’ (((β„‚ D𝑛 𝐹)β€˜0) β†Ύ 𝑆) = (𝐹 β†Ύ 𝑆))
4237, 41eqtr4d 2775 . . . . . 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 11205 . . . . . . . . . 10 β„‚ ∈ {ℝ, β„‚}
45 simplr 767 . . . . . . . . . 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 769 . . . . . . . . . 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 25669 . . . . . . . . . 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 1466 . . . . . . . . 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 771 . . . . . . . . . . 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 4005 . . . . . . . . . . . . 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 25666 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 5905 . . . . . . . . . . 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 2774 . . . . . . . . . 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 25668 . . . . . . . . . . . 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 1466 . . . . . . . . . . 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 11193 . . . . . . . . . . . . . . 15 β„‚ ∈ V
5857, 57elpm2 8870 . . . . . . . . . . . . . 14 (𝐹 ∈ (β„‚ ↑pm β„‚) ↔ (𝐹:dom πΉβŸΆβ„‚ ∧ dom 𝐹 βŠ† β„‚))
5958simprbi 497 . . . . . . . . . . . . 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 3992 . . . . . . . . . . 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 25642 . . . . . . . . . 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 4020 . . . . . . . . 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 3999 . . . . . . . 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 457 . . . . . . 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 7419 . . . . . . 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 481 . . . . . . . . 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 481 . . . . . . . . 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 25666 . . . . . . . . 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 1371 . . . . . . . 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 5980 . . . . . . . . 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 765 . . . . . . . . . 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 2732 . . . . . . . . . . . . . 14 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
7574cnfldtop 24520 . . . . . . . . . . . . 13 (TopOpenβ€˜β„‚fld) ∈ Top
76 unicntop 24522 . . . . . . . . . . . . . 14 β„‚ = βˆͺ (TopOpenβ€˜β„‚fld)
7776ntrss2 22781 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 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 24519 . . . . . . . . . . . . . . . 16 (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚)
8079toponrestid 22643 . . . . . . . . . . . . . . 15 (TopOpenβ€˜β„‚fld) = ((TopOpenβ€˜β„‚fld) β†Ύt β„‚)
8150, 56, 61, 80, 74dvbssntr 25641 . . . . . . . . . . . . . 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 4020 . . . . . . . . . . . . 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 3992 . . . . . . . . . . . 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 3999 . . . . . . . . . . 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 22790 . . . . . . . . . . . 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 587 . . . . . . . . . . 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 256 . . . . . . . . . 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 2773 . . . . . . . . . 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 25655 . . . . . . . . . 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 837 . . . . . . . . 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 2775 . . . . . . . 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 2748 . . . . . . 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 844 . . . . 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 12661 . . . 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 1117 . 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 407 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  {cpr 4630  dom cdm 5676   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ↑pm cpm 8823  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115  β„•0cn0 12476  TopOpenctopn 17371  β„‚fldccnfld 21144  Topctop 22615  intcnt 22741   D cdv 25604   D𝑛 cdvn 25605
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-icc 13335  df-fz 13489  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-struct 17084  df-slot 17119  df-ndx 17131  df-base 17149  df-plusg 17214  df-mulr 17215  df-starv 17216  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-rest 17372  df-topn 17373  df-topgen 17393  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-fbas 21141  df-fg 21142  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-lp 22860  df-perf 22861  df-cnp 22952  df-haus 23039  df-fil 23570  df-fm 23662  df-flim 23663  df-flf 23664  df-xms 24046  df-ms 24047  df-limc 25607  df-dv 25608  df-dvn 25609
This theorem is referenced by:  cpnres  25678
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