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Theorem sylow1lem1 19468
Description: Lemma for sylow1 19473. The p-adic valuation of the size of 𝑆 is equal to the number of excess powers of 𝑃 in (β™―β€˜π‘‹) / (𝑃↑𝑁). (Contributed by Mario Carneiro, 15-Jan-2015.)
Hypotheses
Ref Expression
sylow1.x 𝑋 = (Baseβ€˜πΊ)
sylow1.g (πœ‘ β†’ 𝐺 ∈ Grp)
sylow1.f (πœ‘ β†’ 𝑋 ∈ Fin)
sylow1.p (πœ‘ β†’ 𝑃 ∈ β„™)
sylow1.n (πœ‘ β†’ 𝑁 ∈ β„•0)
sylow1.d (πœ‘ β†’ (𝑃↑𝑁) βˆ₯ (β™―β€˜π‘‹))
sylow1lem.a + = (+gβ€˜πΊ)
sylow1lem.s 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (β™―β€˜π‘ ) = (𝑃↑𝑁)}
Assertion
Ref Expression
sylow1lem1 (πœ‘ β†’ ((β™―β€˜π‘†) ∈ β„• ∧ (𝑃 pCnt (β™―β€˜π‘†)) = ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁)))
Distinct variable groups:   𝑁,𝑠   𝑋,𝑠   + ,𝑠   𝐺,𝑠   𝑃,𝑠
Allowed substitution hints:   πœ‘(𝑠)   𝑆(𝑠)

Proof of Theorem sylow1lem1
Dummy variables π‘₯ 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sylow1.f . . . . 5 (πœ‘ β†’ 𝑋 ∈ Fin)
2 sylow1.p . . . . . . . 8 (πœ‘ β†’ 𝑃 ∈ β„™)
3 prmnn 16613 . . . . . . . 8 (𝑃 ∈ β„™ β†’ 𝑃 ∈ β„•)
42, 3syl 17 . . . . . . 7 (πœ‘ β†’ 𝑃 ∈ β„•)
5 sylow1.n . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ β„•0)
64, 5nnexpcld 14210 . . . . . 6 (πœ‘ β†’ (𝑃↑𝑁) ∈ β„•)
76nnzd 12587 . . . . 5 (πœ‘ β†’ (𝑃↑𝑁) ∈ β„€)
8 hashbc 14414 . . . . 5 ((𝑋 ∈ Fin ∧ (𝑃↑𝑁) ∈ β„€) β†’ ((β™―β€˜π‘‹)C(𝑃↑𝑁)) = (β™―β€˜{𝑠 ∈ 𝒫 𝑋 ∣ (β™―β€˜π‘ ) = (𝑃↑𝑁)}))
91, 7, 8syl2anc 584 . . . 4 (πœ‘ β†’ ((β™―β€˜π‘‹)C(𝑃↑𝑁)) = (β™―β€˜{𝑠 ∈ 𝒫 𝑋 ∣ (β™―β€˜π‘ ) = (𝑃↑𝑁)}))
10 sylow1lem.s . . . . 5 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (β™―β€˜π‘ ) = (𝑃↑𝑁)}
1110fveq2i 6894 . . . 4 (β™―β€˜π‘†) = (β™―β€˜{𝑠 ∈ 𝒫 𝑋 ∣ (β™―β€˜π‘ ) = (𝑃↑𝑁)})
129, 11eqtr4di 2790 . . 3 (πœ‘ β†’ ((β™―β€˜π‘‹)C(𝑃↑𝑁)) = (β™―β€˜π‘†))
13 sylow1.d . . . . . 6 (πœ‘ β†’ (𝑃↑𝑁) βˆ₯ (β™―β€˜π‘‹))
14 sylow1.g . . . . . . . . . 10 (πœ‘ β†’ 𝐺 ∈ Grp)
15 sylow1.x . . . . . . . . . . 11 𝑋 = (Baseβ€˜πΊ)
1615grpbn0 18853 . . . . . . . . . 10 (𝐺 ∈ Grp β†’ 𝑋 β‰  βˆ…)
1714, 16syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝑋 β‰  βˆ…)
18 hasheq0 14325 . . . . . . . . . . 11 (𝑋 ∈ Fin β†’ ((β™―β€˜π‘‹) = 0 ↔ 𝑋 = βˆ…))
191, 18syl 17 . . . . . . . . . 10 (πœ‘ β†’ ((β™―β€˜π‘‹) = 0 ↔ 𝑋 = βˆ…))
2019necon3bbid 2978 . . . . . . . . 9 (πœ‘ β†’ (Β¬ (β™―β€˜π‘‹) = 0 ↔ 𝑋 β‰  βˆ…))
2117, 20mpbird 256 . . . . . . . 8 (πœ‘ β†’ Β¬ (β™―β€˜π‘‹) = 0)
22 hashcl 14318 . . . . . . . . . . 11 (𝑋 ∈ Fin β†’ (β™―β€˜π‘‹) ∈ β„•0)
231, 22syl 17 . . . . . . . . . 10 (πœ‘ β†’ (β™―β€˜π‘‹) ∈ β„•0)
24 elnn0 12476 . . . . . . . . . 10 ((β™―β€˜π‘‹) ∈ β„•0 ↔ ((β™―β€˜π‘‹) ∈ β„• ∨ (β™―β€˜π‘‹) = 0))
2523, 24sylib 217 . . . . . . . . 9 (πœ‘ β†’ ((β™―β€˜π‘‹) ∈ β„• ∨ (β™―β€˜π‘‹) = 0))
2625ord 862 . . . . . . . 8 (πœ‘ β†’ (Β¬ (β™―β€˜π‘‹) ∈ β„• β†’ (β™―β€˜π‘‹) = 0))
2721, 26mt3d 148 . . . . . . 7 (πœ‘ β†’ (β™―β€˜π‘‹) ∈ β„•)
28 dvdsle 16255 . . . . . . 7 (((𝑃↑𝑁) ∈ β„€ ∧ (β™―β€˜π‘‹) ∈ β„•) β†’ ((𝑃↑𝑁) βˆ₯ (β™―β€˜π‘‹) β†’ (𝑃↑𝑁) ≀ (β™―β€˜π‘‹)))
297, 27, 28syl2anc 584 . . . . . 6 (πœ‘ β†’ ((𝑃↑𝑁) βˆ₯ (β™―β€˜π‘‹) β†’ (𝑃↑𝑁) ≀ (β™―β€˜π‘‹)))
3013, 29mpd 15 . . . . 5 (πœ‘ β†’ (𝑃↑𝑁) ≀ (β™―β€˜π‘‹))
316nnnn0d 12534 . . . . . . 7 (πœ‘ β†’ (𝑃↑𝑁) ∈ β„•0)
32 nn0uz 12866 . . . . . . 7 β„•0 = (β„€β‰₯β€˜0)
3331, 32eleqtrdi 2843 . . . . . 6 (πœ‘ β†’ (𝑃↑𝑁) ∈ (β„€β‰₯β€˜0))
3423nn0zd 12586 . . . . . 6 (πœ‘ β†’ (β™―β€˜π‘‹) ∈ β„€)
35 elfz5 13495 . . . . . 6 (((𝑃↑𝑁) ∈ (β„€β‰₯β€˜0) ∧ (β™―β€˜π‘‹) ∈ β„€) β†’ ((𝑃↑𝑁) ∈ (0...(β™―β€˜π‘‹)) ↔ (𝑃↑𝑁) ≀ (β™―β€˜π‘‹)))
3633, 34, 35syl2anc 584 . . . . 5 (πœ‘ β†’ ((𝑃↑𝑁) ∈ (0...(β™―β€˜π‘‹)) ↔ (𝑃↑𝑁) ≀ (β™―β€˜π‘‹)))
3730, 36mpbird 256 . . . 4 (πœ‘ β†’ (𝑃↑𝑁) ∈ (0...(β™―β€˜π‘‹)))
38 bccl2 14285 . . . 4 ((𝑃↑𝑁) ∈ (0...(β™―β€˜π‘‹)) β†’ ((β™―β€˜π‘‹)C(𝑃↑𝑁)) ∈ β„•)
3937, 38syl 17 . . 3 (πœ‘ β†’ ((β™―β€˜π‘‹)C(𝑃↑𝑁)) ∈ β„•)
4012, 39eqeltrrd 2834 . 2 (πœ‘ β†’ (β™―β€˜π‘†) ∈ β„•)
41 nnuz 12867 . . . . . . . . . . 11 β„• = (β„€β‰₯β€˜1)
426, 41eleqtrdi 2843 . . . . . . . . . 10 (πœ‘ β†’ (𝑃↑𝑁) ∈ (β„€β‰₯β€˜1))
43 elfz5 13495 . . . . . . . . . 10 (((𝑃↑𝑁) ∈ (β„€β‰₯β€˜1) ∧ (β™―β€˜π‘‹) ∈ β„€) β†’ ((𝑃↑𝑁) ∈ (1...(β™―β€˜π‘‹)) ↔ (𝑃↑𝑁) ≀ (β™―β€˜π‘‹)))
4442, 34, 43syl2anc 584 . . . . . . . . 9 (πœ‘ β†’ ((𝑃↑𝑁) ∈ (1...(β™―β€˜π‘‹)) ↔ (𝑃↑𝑁) ≀ (β™―β€˜π‘‹)))
4530, 44mpbird 256 . . . . . . . 8 (πœ‘ β†’ (𝑃↑𝑁) ∈ (1...(β™―β€˜π‘‹)))
46 1zzd 12595 . . . . . . . . 9 (πœ‘ β†’ 1 ∈ β„€)
47 fzsubel 13539 . . . . . . . . 9 (((1 ∈ β„€ ∧ (β™―β€˜π‘‹) ∈ β„€) ∧ ((𝑃↑𝑁) ∈ β„€ ∧ 1 ∈ β„€)) β†’ ((𝑃↑𝑁) ∈ (1...(β™―β€˜π‘‹)) ↔ ((𝑃↑𝑁) βˆ’ 1) ∈ ((1 βˆ’ 1)...((β™―β€˜π‘‹) βˆ’ 1))))
4846, 34, 7, 46, 47syl22anc 837 . . . . . . . 8 (πœ‘ β†’ ((𝑃↑𝑁) ∈ (1...(β™―β€˜π‘‹)) ↔ ((𝑃↑𝑁) βˆ’ 1) ∈ ((1 βˆ’ 1)...((β™―β€˜π‘‹) βˆ’ 1))))
4945, 48mpbid 231 . . . . . . 7 (πœ‘ β†’ ((𝑃↑𝑁) βˆ’ 1) ∈ ((1 βˆ’ 1)...((β™―β€˜π‘‹) βˆ’ 1)))
50 1m1e0 12286 . . . . . . . 8 (1 βˆ’ 1) = 0
5150oveq1i 7421 . . . . . . 7 ((1 βˆ’ 1)...((β™―β€˜π‘‹) βˆ’ 1)) = (0...((β™―β€˜π‘‹) βˆ’ 1))
5249, 51eleqtrdi 2843 . . . . . 6 (πœ‘ β†’ ((𝑃↑𝑁) βˆ’ 1) ∈ (0...((β™―β€˜π‘‹) βˆ’ 1)))
53 bcp1nk 14279 . . . . . 6 (((𝑃↑𝑁) βˆ’ 1) ∈ (0...((β™―β€˜π‘‹) βˆ’ 1)) β†’ ((((β™―β€˜π‘‹) βˆ’ 1) + 1)C(((𝑃↑𝑁) βˆ’ 1) + 1)) = ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) Β· ((((β™―β€˜π‘‹) βˆ’ 1) + 1) / (((𝑃↑𝑁) βˆ’ 1) + 1))))
5452, 53syl 17 . . . . 5 (πœ‘ β†’ ((((β™―β€˜π‘‹) βˆ’ 1) + 1)C(((𝑃↑𝑁) βˆ’ 1) + 1)) = ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) Β· ((((β™―β€˜π‘‹) βˆ’ 1) + 1) / (((𝑃↑𝑁) βˆ’ 1) + 1))))
5523nn0cnd 12536 . . . . . . 7 (πœ‘ β†’ (β™―β€˜π‘‹) ∈ β„‚)
56 ax-1cn 11170 . . . . . . 7 1 ∈ β„‚
57 npcan 11471 . . . . . . 7 (((β™―β€˜π‘‹) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ (((β™―β€˜π‘‹) βˆ’ 1) + 1) = (β™―β€˜π‘‹))
5855, 56, 57sylancl 586 . . . . . 6 (πœ‘ β†’ (((β™―β€˜π‘‹) βˆ’ 1) + 1) = (β™―β€˜π‘‹))
596nncnd 12230 . . . . . . 7 (πœ‘ β†’ (𝑃↑𝑁) ∈ β„‚)
60 npcan 11471 . . . . . . 7 (((𝑃↑𝑁) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ (((𝑃↑𝑁) βˆ’ 1) + 1) = (𝑃↑𝑁))
6159, 56, 60sylancl 586 . . . . . 6 (πœ‘ β†’ (((𝑃↑𝑁) βˆ’ 1) + 1) = (𝑃↑𝑁))
6258, 61oveq12d 7429 . . . . 5 (πœ‘ β†’ ((((β™―β€˜π‘‹) βˆ’ 1) + 1)C(((𝑃↑𝑁) βˆ’ 1) + 1)) = ((β™―β€˜π‘‹)C(𝑃↑𝑁)))
6358, 61oveq12d 7429 . . . . . 6 (πœ‘ β†’ ((((β™―β€˜π‘‹) βˆ’ 1) + 1) / (((𝑃↑𝑁) βˆ’ 1) + 1)) = ((β™―β€˜π‘‹) / (𝑃↑𝑁)))
6463oveq2d 7427 . . . . 5 (πœ‘ β†’ ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) Β· ((((β™―β€˜π‘‹) βˆ’ 1) + 1) / (((𝑃↑𝑁) βˆ’ 1) + 1))) = ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) Β· ((β™―β€˜π‘‹) / (𝑃↑𝑁))))
6554, 62, 643eqtr3d 2780 . . . 4 (πœ‘ β†’ ((β™―β€˜π‘‹)C(𝑃↑𝑁)) = ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) Β· ((β™―β€˜π‘‹) / (𝑃↑𝑁))))
6665oveq2d 7427 . . 3 (πœ‘ β†’ (𝑃 pCnt ((β™―β€˜π‘‹)C(𝑃↑𝑁))) = (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) Β· ((β™―β€˜π‘‹) / (𝑃↑𝑁)))))
6712oveq2d 7427 . . 3 (πœ‘ β†’ (𝑃 pCnt ((β™―β€˜π‘‹)C(𝑃↑𝑁))) = (𝑃 pCnt (β™―β€˜π‘†)))
68 bccl2 14285 . . . . . . 7 (((𝑃↑𝑁) βˆ’ 1) ∈ (0...((β™―β€˜π‘‹) βˆ’ 1)) β†’ (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) ∈ β„•)
6952, 68syl 17 . . . . . 6 (πœ‘ β†’ (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) ∈ β„•)
7069nnzd 12587 . . . . 5 (πœ‘ β†’ (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) ∈ β„€)
7169nnne0d 12264 . . . . 5 (πœ‘ β†’ (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) β‰  0)
726nnne0d 12264 . . . . . . 7 (πœ‘ β†’ (𝑃↑𝑁) β‰  0)
73 dvdsval2 16202 . . . . . . 7 (((𝑃↑𝑁) ∈ β„€ ∧ (𝑃↑𝑁) β‰  0 ∧ (β™―β€˜π‘‹) ∈ β„€) β†’ ((𝑃↑𝑁) βˆ₯ (β™―β€˜π‘‹) ↔ ((β™―β€˜π‘‹) / (𝑃↑𝑁)) ∈ β„€))
747, 72, 34, 73syl3anc 1371 . . . . . 6 (πœ‘ β†’ ((𝑃↑𝑁) βˆ₯ (β™―β€˜π‘‹) ↔ ((β™―β€˜π‘‹) / (𝑃↑𝑁)) ∈ β„€))
7513, 74mpbid 231 . . . . 5 (πœ‘ β†’ ((β™―β€˜π‘‹) / (𝑃↑𝑁)) ∈ β„€)
7627nnne0d 12264 . . . . . 6 (πœ‘ β†’ (β™―β€˜π‘‹) β‰  0)
7755, 59, 76, 72divne0d 12008 . . . . 5 (πœ‘ β†’ ((β™―β€˜π‘‹) / (𝑃↑𝑁)) β‰  0)
78 pcmul 16786 . . . . 5 ((𝑃 ∈ β„™ ∧ ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) ∈ β„€ ∧ (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) β‰  0) ∧ (((β™―β€˜π‘‹) / (𝑃↑𝑁)) ∈ β„€ ∧ ((β™―β€˜π‘‹) / (𝑃↑𝑁)) β‰  0)) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) Β· ((β™―β€˜π‘‹) / (𝑃↑𝑁)))) = ((𝑃 pCnt (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1))) + (𝑃 pCnt ((β™―β€˜π‘‹) / (𝑃↑𝑁)))))
792, 70, 71, 75, 77, 78syl122anc 1379 . . . 4 (πœ‘ β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) Β· ((β™―β€˜π‘‹) / (𝑃↑𝑁)))) = ((𝑃 pCnt (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1))) + (𝑃 pCnt ((β™―β€˜π‘‹) / (𝑃↑𝑁)))))
80 1cnd 11211 . . . . . . . . 9 (πœ‘ β†’ 1 ∈ β„‚)
8155, 59, 80npncand 11597 . . . . . . . 8 (πœ‘ β†’ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1)) = ((β™―β€˜π‘‹) βˆ’ 1))
8281oveq1d 7426 . . . . . . 7 (πœ‘ β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1)) = (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)))
8382oveq2d 7427 . . . . . 6 (πœ‘ β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1))) = (𝑃 pCnt (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1))))
846nnred 12229 . . . . . . . 8 (πœ‘ β†’ (𝑃↑𝑁) ∈ ℝ)
8584ltm1d 12148 . . . . . . 7 (πœ‘ β†’ ((𝑃↑𝑁) βˆ’ 1) < (𝑃↑𝑁))
86 nnm1nn0 12515 . . . . . . . . 9 ((𝑃↑𝑁) ∈ β„• β†’ ((𝑃↑𝑁) βˆ’ 1) ∈ β„•0)
876, 86syl 17 . . . . . . . 8 (πœ‘ β†’ ((𝑃↑𝑁) βˆ’ 1) ∈ β„•0)
88 breq1 5151 . . . . . . . . . . 11 (π‘₯ = 0 β†’ (π‘₯ < (𝑃↑𝑁) ↔ 0 < (𝑃↑𝑁)))
89 bcxmaslem1 15782 . . . . . . . . . . . . 13 (π‘₯ = 0 β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯) = ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0))
9089oveq2d 7427 . . . . . . . . . . . 12 (π‘₯ = 0 β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0)))
9190eqeq1d 2734 . . . . . . . . . . 11 (π‘₯ = 0 β†’ ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0 ↔ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0)) = 0))
9288, 91imbi12d 344 . . . . . . . . . 10 (π‘₯ = 0 β†’ ((π‘₯ < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0) ↔ (0 < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0)) = 0)))
9392imbi2d 340 . . . . . . . . 9 (π‘₯ = 0 β†’ ((πœ‘ β†’ (π‘₯ < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0)) ↔ (πœ‘ β†’ (0 < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0)) = 0))))
94 breq1 5151 . . . . . . . . . . 11 (π‘₯ = 𝑛 β†’ (π‘₯ < (𝑃↑𝑁) ↔ 𝑛 < (𝑃↑𝑁)))
95 bcxmaslem1 15782 . . . . . . . . . . . . 13 (π‘₯ = 𝑛 β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯) = ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛))
9695oveq2d 7427 . . . . . . . . . . . 12 (π‘₯ = 𝑛 β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)))
9796eqeq1d 2734 . . . . . . . . . . 11 (π‘₯ = 𝑛 β†’ ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0 ↔ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0))
9894, 97imbi12d 344 . . . . . . . . . 10 (π‘₯ = 𝑛 β†’ ((π‘₯ < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0) ↔ (𝑛 < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0)))
9998imbi2d 340 . . . . . . . . 9 (π‘₯ = 𝑛 β†’ ((πœ‘ β†’ (π‘₯ < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0)) ↔ (πœ‘ β†’ (𝑛 < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0))))
100 breq1 5151 . . . . . . . . . . 11 (π‘₯ = (𝑛 + 1) β†’ (π‘₯ < (𝑃↑𝑁) ↔ (𝑛 + 1) < (𝑃↑𝑁)))
101 bcxmaslem1 15782 . . . . . . . . . . . . 13 (π‘₯ = (𝑛 + 1) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯) = ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
102101oveq2d 7427 . . . . . . . . . . . 12 (π‘₯ = (𝑛 + 1) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))))
103102eqeq1d 2734 . . . . . . . . . . 11 (π‘₯ = (𝑛 + 1) β†’ ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0 ↔ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))
104100, 103imbi12d 344 . . . . . . . . . 10 (π‘₯ = (𝑛 + 1) β†’ ((π‘₯ < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0) ↔ ((𝑛 + 1) < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0)))
105104imbi2d 340 . . . . . . . . 9 (π‘₯ = (𝑛 + 1) β†’ ((πœ‘ β†’ (π‘₯ < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0)) ↔ (πœ‘ β†’ ((𝑛 + 1) < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))))
106 breq1 5151 . . . . . . . . . . 11 (π‘₯ = ((𝑃↑𝑁) βˆ’ 1) β†’ (π‘₯ < (𝑃↑𝑁) ↔ ((𝑃↑𝑁) βˆ’ 1) < (𝑃↑𝑁)))
107 bcxmaslem1 15782 . . . . . . . . . . . . 13 (π‘₯ = ((𝑃↑𝑁) βˆ’ 1) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯) = ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1)))
108107oveq2d 7427 . . . . . . . . . . . 12 (π‘₯ = ((𝑃↑𝑁) βˆ’ 1) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1))))
109108eqeq1d 2734 . . . . . . . . . . 11 (π‘₯ = ((𝑃↑𝑁) βˆ’ 1) β†’ ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0 ↔ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1))) = 0))
110106, 109imbi12d 344 . . . . . . . . . 10 (π‘₯ = ((𝑃↑𝑁) βˆ’ 1) β†’ ((π‘₯ < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0) ↔ (((𝑃↑𝑁) βˆ’ 1) < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1))) = 0)))
111110imbi2d 340 . . . . . . . . 9 (π‘₯ = ((𝑃↑𝑁) βˆ’ 1) β†’ ((πœ‘ β†’ (π‘₯ < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + π‘₯)Cπ‘₯)) = 0)) ↔ (πœ‘ β†’ (((𝑃↑𝑁) βˆ’ 1) < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1))) = 0))))
112 znn0sub 12611 . . . . . . . . . . . . . . . 16 (((𝑃↑𝑁) ∈ β„€ ∧ (β™―β€˜π‘‹) ∈ β„€) β†’ ((𝑃↑𝑁) ≀ (β™―β€˜π‘‹) ↔ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„•0))
1137, 34, 112syl2anc 584 . . . . . . . . . . . . . . 15 (πœ‘ β†’ ((𝑃↑𝑁) ≀ (β™―β€˜π‘‹) ↔ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„•0))
11430, 113mpbid 231 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„•0)
115 0nn0 12489 . . . . . . . . . . . . . 14 0 ∈ β„•0
116 nn0addcl 12509 . . . . . . . . . . . . . 14 ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„•0 ∧ 0 ∈ β„•0) β†’ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0) ∈ β„•0)
117114, 115, 116sylancl 586 . . . . . . . . . . . . 13 (πœ‘ β†’ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0) ∈ β„•0)
118 bcn0 14272 . . . . . . . . . . . . 13 ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0) ∈ β„•0 β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0) = 1)
119117, 118syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0) = 1)
120119oveq2d 7427 . . . . . . . . . . 11 (πœ‘ β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0)) = (𝑃 pCnt 1))
121 pc1 16790 . . . . . . . . . . . 12 (𝑃 ∈ β„™ β†’ (𝑃 pCnt 1) = 0)
1222, 121syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (𝑃 pCnt 1) = 0)
123120, 122eqtrd 2772 . . . . . . . . . 10 (πœ‘ β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0)) = 0)
124123a1d 25 . . . . . . . . 9 (πœ‘ β†’ (0 < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 0)C0)) = 0))
125 nn0re 12483 . . . . . . . . . . . . . 14 (𝑛 ∈ β„•0 β†’ 𝑛 ∈ ℝ)
126125ad2antrl 726 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑛 ∈ ℝ)
127 nn0p1nn 12513 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„•0 β†’ (𝑛 + 1) ∈ β„•)
128127ad2antrl 726 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑛 + 1) ∈ β„•)
129128nnred 12229 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑛 + 1) ∈ ℝ)
1306adantr 481 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃↑𝑁) ∈ β„•)
131130nnred 12229 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃↑𝑁) ∈ ℝ)
132126ltp1d 12146 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑛 < (𝑛 + 1))
133 simprr 771 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑛 + 1) < (𝑃↑𝑁))
134126, 129, 131, 132, 133lttrd 11377 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑛 < (𝑃↑𝑁))
135134expr 457 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 + 1) < (𝑃↑𝑁) β†’ 𝑛 < (𝑃↑𝑁)))
136135imim1d 82 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0) β†’ ((𝑛 + 1) < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0)))
137 oveq1 7418 . . . . . . . . . . 11 ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0 β†’ ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
138114adantr 481 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„•0)
139138nn0cnd 12536 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„‚)
140 nn0cn 12484 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ β„•0 β†’ 𝑛 ∈ β„‚)
141140ad2antrl 726 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑛 ∈ β„‚)
142 1cnd 11211 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 1 ∈ β„‚)
143139, 141, 142addassd 11238 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) = (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1)))
144143oveq1d 7426 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
145 nn0addge2 12521 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℝ ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„•0) β†’ 𝑛 ≀ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛))
146126, 138, 145syl2anc 584 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑛 ≀ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛))
147 simprl 769 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑛 ∈ β„•0)
148147, 32eleqtrdi 2843 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑛 ∈ (β„€β‰₯β€˜0))
149138, 147nn0addcld 12538 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) ∈ β„•0)
150149nn0zd 12586 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) ∈ β„€)
151 elfz5 13495 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (β„€β‰₯β€˜0) ∧ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) ∈ β„€) β†’ (𝑛 ∈ (0...(((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)) ↔ 𝑛 ≀ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)))
152148, 150, 151syl2anc 584 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑛 ∈ (0...(((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)) ↔ 𝑛 ≀ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)))
153146, 152mpbird 256 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑛 ∈ (0...(((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)))
154 bcp1nk 14279 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (0...(((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)) β†’ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) Β· (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))))
155153, 154syl 17 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) Β· (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))))
156144, 155eqtr3d 2774 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1)) = (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) Β· (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))))
157156oveq2d 7427 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) Β· (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
1582adantr 481 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑃 ∈ β„™)
159 bccl2 14285 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (0...(((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ β„•)
160153, 159syl 17 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ β„•)
161 nnq 12948 . . . . . . . . . . . . . . 15 (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ β„• β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ β„š)
162160, 161syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ β„š)
163160nnne0d 12264 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) β‰  0)
164150peano2zd 12671 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) ∈ β„€)
165 znq 12938 . . . . . . . . . . . . . . 15 ((((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) ∈ β„€ ∧ (𝑛 + 1) ∈ β„•) β†’ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ β„š)
166164, 128, 165syl2anc 584 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ β„š)
167 nn0p1nn 12513 . . . . . . . . . . . . . . . . 17 ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) ∈ β„•0 β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) ∈ β„•)
168149, 167syl 17 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) ∈ β„•)
169 nnrp 12987 . . . . . . . . . . . . . . . . 17 (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) ∈ β„• β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℝ+)
170 nnrp 12987 . . . . . . . . . . . . . . . . 17 ((𝑛 + 1) ∈ β„• β†’ (𝑛 + 1) ∈ ℝ+)
171 rpdivcl 13001 . . . . . . . . . . . . . . . . 17 ((((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℝ+ ∧ (𝑛 + 1) ∈ ℝ+) β†’ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
172169, 170, 171syl2an 596 . . . . . . . . . . . . . . . 16 ((((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) ∈ β„• ∧ (𝑛 + 1) ∈ β„•) β†’ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
173168, 128, 172syl2anc 584 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
174173rpne0d 13023 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) β‰  0)
175 pcqmul 16788 . . . . . . . . . . . . . 14 ((𝑃 ∈ β„™ ∧ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ β„š ∧ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) β‰  0) ∧ ((((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ β„š ∧ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) β‰  0)) β†’ (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) Β· (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
176158, 162, 163, 166, 174, 175syl122anc 1379 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛) Β· (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
177157, 176eqtrd 2772 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
178168nnne0d 12264 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) β‰  0)
179 pcdiv 16787 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ β„™ ∧ (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) ∈ β„€ ∧ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) β‰  0) ∧ (𝑛 + 1) ∈ β„•) β†’ (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1)) βˆ’ (𝑃 pCnt (𝑛 + 1))))
180158, 164, 178, 128, 179syl121anc 1375 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1)) βˆ’ (𝑃 pCnt (𝑛 + 1))))
181128nncnd 12230 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑛 + 1) ∈ β„‚)
182139, 181, 143comraddd 11430 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) = ((𝑛 + 1) + ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))))
183182oveq2d 7427 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)))))
184 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0)
185184oveq2d 7427 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0) β†’ ((𝑛 + 1) + ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))) = ((𝑛 + 1) + 0))
186181addridd 11416 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((𝑛 + 1) + 0) = (𝑛 + 1))
187186adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0) β†’ ((𝑛 + 1) + 0) = (𝑛 + 1))
188185, 187eqtr2d 2773 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0) β†’ (𝑛 + 1) = ((𝑛 + 1) + ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))))
189188oveq2d 7427 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0) β†’ (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)))))
1902ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ 𝑃 ∈ β„™)
191 nnq 12948 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 + 1) ∈ β„• β†’ (𝑛 + 1) ∈ β„š)
192128, 191syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑛 + 1) ∈ β„š)
193192adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (𝑛 + 1) ∈ β„š)
194138nn0zd 12586 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„€)
195 zq 12940 . . . . . . . . . . . . . . . . . . . . . 22 (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„€ β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„š)
196194, 195syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„š)
197196adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„š)
198158, 128pccld 16785 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt (𝑛 + 1)) ∈ β„•0)
199198nn0red 12535 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
200199adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
2015adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑁 ∈ β„•0)
202201nn0red 12535 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 𝑁 ∈ ℝ)
203202adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ 𝑁 ∈ ℝ)
204 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0)
205204neneqd 2945 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ Β¬ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0)
206114ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„•0)
207 elnn0 12476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„•0 ↔ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„• ∨ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0))
208206, 207sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„• ∨ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0))
209208ord 862 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (Β¬ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„• β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) = 0))
210205, 209mt3d 148 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„•)
211190, 210pccld 16785 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (𝑃 pCnt ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))) ∈ β„•0)
212211nn0red 12535 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (𝑃 pCnt ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))) ∈ ℝ)
213128nnzd 12587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑛 + 1) ∈ β„€)
214 pcdvdsb 16804 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ β„™ ∧ (𝑛 + 1) ∈ β„€ ∧ 𝑁 ∈ β„•0) β†’ (𝑁 ≀ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑁) βˆ₯ (𝑛 + 1)))
215158, 213, 201, 214syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑁 ≀ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑁) βˆ₯ (𝑛 + 1)))
2167adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃↑𝑁) ∈ β„€)
217 dvdsle 16255 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃↑𝑁) ∈ β„€ ∧ (𝑛 + 1) ∈ β„•) β†’ ((𝑃↑𝑁) βˆ₯ (𝑛 + 1) β†’ (𝑃↑𝑁) ≀ (𝑛 + 1)))
218216, 128, 217syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((𝑃↑𝑁) βˆ₯ (𝑛 + 1) β†’ (𝑃↑𝑁) ≀ (𝑛 + 1)))
219215, 218sylbid 239 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑁 ≀ (𝑃 pCnt (𝑛 + 1)) β†’ (𝑃↑𝑁) ≀ (𝑛 + 1)))
220202, 199lenltd 11362 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑁 ≀ (𝑃 pCnt (𝑛 + 1)) ↔ Β¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁))
221131, 129lenltd 11362 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((𝑃↑𝑁) ≀ (𝑛 + 1) ↔ Β¬ (𝑛 + 1) < (𝑃↑𝑁)))
222219, 220, 2213imtr3d 292 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (Β¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁 β†’ Β¬ (𝑛 + 1) < (𝑃↑𝑁)))
223133, 222mt4d 117 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt (𝑛 + 1)) < 𝑁)
224223adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (𝑃 pCnt (𝑛 + 1)) < 𝑁)
225 dvdssubr 16250 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃↑𝑁) ∈ β„€ ∧ (β™―β€˜π‘‹) ∈ β„€) β†’ ((𝑃↑𝑁) βˆ₯ (β™―β€˜π‘‹) ↔ (𝑃↑𝑁) βˆ₯ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))))
2267, 34, 225syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 (πœ‘ β†’ ((𝑃↑𝑁) βˆ₯ (β™―β€˜π‘‹) ↔ (𝑃↑𝑁) βˆ₯ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))))
22713, 226mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ (𝑃↑𝑁) βˆ₯ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)))
228227ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (𝑃↑𝑁) βˆ₯ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)))
229206nn0zd 12586 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„€)
2305ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ 𝑁 ∈ β„•0)
231 pcdvdsb 16804 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ β„™ ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) ∈ β„€ ∧ 𝑁 ∈ β„•0) β†’ (𝑁 ≀ (𝑃 pCnt ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))) ↔ (𝑃↑𝑁) βˆ₯ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))))
232190, 229, 230, 231syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (𝑁 ≀ (𝑃 pCnt ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))) ↔ (𝑃↑𝑁) βˆ₯ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))))
233228, 232mpbird 256 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ 𝑁 ≀ (𝑃 pCnt ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))))
234200, 203, 212, 224, 233ltletrd 11376 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (𝑃 pCnt (𝑛 + 1)) < (𝑃 pCnt ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁))))
235190, 193, 197, 234pcadd2 16825 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) β‰  0) β†’ (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)))))
236189, 235pm2.61dane 3029 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)))))
237183, 236eqtr4d 2775 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1)) = (𝑃 pCnt (𝑛 + 1)))
238198nn0cnd 12536 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt (𝑛 + 1)) ∈ β„‚)
239237, 238eqeltrd 2833 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1)) ∈ β„‚)
240239, 237subeq0bd 11642 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1)) βˆ’ (𝑃 pCnt (𝑛 + 1))) = 0)
241180, 240eqtrd 2772 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = 0)
242241oveq2d 7427 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ (0 + (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + 0))
243 00id 11391 . . . . . . . . . . . . 13 (0 + 0) = 0
244242, 243eqtr2di 2789 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ 0 = (0 + (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
245177, 244eqeq12d 2748 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0 ↔ ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + (𝑃 pCnt (((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))))))
246137, 245imbitrrid 245 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 ∈ β„•0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) β†’ ((𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0 β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))
247136, 246animpimp2impd 844 . . . . . . . . 9 (𝑛 ∈ β„•0 β†’ ((πœ‘ β†’ (𝑛 < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0)) β†’ (πœ‘ β†’ ((𝑛 + 1) < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))))
24893, 99, 105, 111, 124, 247nn0ind 12659 . . . . . . . 8 (((𝑃↑𝑁) βˆ’ 1) ∈ β„•0 β†’ (πœ‘ β†’ (((𝑃↑𝑁) βˆ’ 1) < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1))) = 0)))
24987, 248mpcom 38 . . . . . . 7 (πœ‘ β†’ (((𝑃↑𝑁) βˆ’ 1) < (𝑃↑𝑁) β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1))) = 0))
25085, 249mpd 15 . . . . . 6 (πœ‘ β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ (𝑃↑𝑁)) + ((𝑃↑𝑁) βˆ’ 1))C((𝑃↑𝑁) βˆ’ 1))) = 0)
25183, 250eqtr3d 2774 . . . . 5 (πœ‘ β†’ (𝑃 pCnt (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1))) = 0)
252 pcdiv 16787 . . . . . . 7 ((𝑃 ∈ β„™ ∧ ((β™―β€˜π‘‹) ∈ β„€ ∧ (β™―β€˜π‘‹) β‰  0) ∧ (𝑃↑𝑁) ∈ β„•) β†’ (𝑃 pCnt ((β™―β€˜π‘‹) / (𝑃↑𝑁))) = ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ (𝑃 pCnt (𝑃↑𝑁))))
2532, 34, 76, 6, 252syl121anc 1375 . . . . . 6 (πœ‘ β†’ (𝑃 pCnt ((β™―β€˜π‘‹) / (𝑃↑𝑁))) = ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ (𝑃 pCnt (𝑃↑𝑁))))
2545nn0zd 12586 . . . . . . . 8 (πœ‘ β†’ 𝑁 ∈ β„€)
255 pcid 16808 . . . . . . . 8 ((𝑃 ∈ β„™ ∧ 𝑁 ∈ β„€) β†’ (𝑃 pCnt (𝑃↑𝑁)) = 𝑁)
2562, 254, 255syl2anc 584 . . . . . . 7 (πœ‘ β†’ (𝑃 pCnt (𝑃↑𝑁)) = 𝑁)
257256oveq2d 7427 . . . . . 6 (πœ‘ β†’ ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ (𝑃 pCnt (𝑃↑𝑁))) = ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁))
258253, 257eqtrd 2772 . . . . 5 (πœ‘ β†’ (𝑃 pCnt ((β™―β€˜π‘‹) / (𝑃↑𝑁))) = ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁))
259251, 258oveq12d 7429 . . . 4 (πœ‘ β†’ ((𝑃 pCnt (((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1))) + (𝑃 pCnt ((β™―β€˜π‘‹) / (𝑃↑𝑁)))) = (0 + ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁)))
2602, 27pccld 16785 . . . . . . . 8 (πœ‘ β†’ (𝑃 pCnt (β™―β€˜π‘‹)) ∈ β„•0)
261260nn0zd 12586 . . . . . . 7 (πœ‘ β†’ (𝑃 pCnt (β™―β€˜π‘‹)) ∈ β„€)
262261, 254zsubcld 12673 . . . . . 6 (πœ‘ β†’ ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁) ∈ β„€)
263262zcnd 12669 . . . . 5 (πœ‘ β†’ ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁) ∈ β„‚)
264263addlidd 11417 . . . 4 (πœ‘ β†’ (0 + ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁)) = ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁))
26579, 259, 2643eqtrd 2776 . . 3 (πœ‘ β†’ (𝑃 pCnt ((((β™―β€˜π‘‹) βˆ’ 1)C((𝑃↑𝑁) βˆ’ 1)) Β· ((β™―β€˜π‘‹) / (𝑃↑𝑁)))) = ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁))
26666, 67, 2653eqtr3d 2780 . 2 (πœ‘ β†’ (𝑃 pCnt (β™―β€˜π‘†)) = ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁))
26740, 266jca 512 1 (πœ‘ β†’ ((β™―β€˜π‘†) ∈ β„• ∧ (𝑃 pCnt (β™―β€˜π‘†)) = ((𝑃 pCnt (β™―β€˜π‘‹)) βˆ’ 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432  βˆ…c0 4322  π’« cpw 4602   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Fincfn 8941  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   Β· cmul 11117   < clt 11250   ≀ cle 11251   βˆ’ cmin 11446   / cdiv 11873  β„•cn 12214  β„•0cn0 12474  β„€cz 12560  β„€β‰₯cuz 12824  β„šcq 12934  β„+crp 12976  ...cfz 13486  β†‘cexp 14029  Ccbc 14264  β™―chash 14292   βˆ₯ cdvds 16199  β„™cprime 16610   pCnt cpc 16771  Basecbs 17146  +gcplusg 17199  Grpcgrp 18821
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-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  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-op 4635  df-uni 4909  df-int 4951  df-iun 4999  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-2o 8469  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-fz 13487  df-fl 13759  df-mod 13837  df-seq 13969  df-exp 14030  df-fac 14236  df-bc 14265  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-dvds 16200  df-gcd 16438  df-prm 16611  df-pc 16772  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-grp 18824
This theorem is referenced by:  sylow1lem3  19470
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