Theorem isprm2lem 15232

Description: Lemma for isprm2 15233. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
isprm2lem	⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃}))

Distinct variable group:   𝑃,𝑛

Proof of Theorem isprm2lem

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 2844	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑝 ≠ 1 ↔ 𝑃 ≠ 1))
2		breq2 4587	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑛 ∥ 𝑝 ↔ 𝑛 ∥ 𝑃))
3	2	rabbidv 3164	. . . . . 6 ⊢ (𝑝 = 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃})
4	3	breq1d 4593	. . . . 5 ⊢ (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2𝑜))
5		preq2 4213	. . . . . 6 ⊢ (𝑝 = 𝑃 → {1, 𝑝} = {1, 𝑃})
6	3, 5	eqeq12d 2625	. . . . 5 ⊢ (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝} ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃}))
7	4, 6	bibi12d 334	. . . 4 ⊢ (𝑝 = 𝑃 → (({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝}) ↔ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃})))
8	1, 7	imbi12d 333	. . 3 ⊢ (𝑝 = 𝑃 → ((𝑝 ≠ 1 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝})) ↔ (𝑃 ≠ 1 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃}))))
9		1idssfct 15231	. . . . . . . 8 ⊢ (𝑝 ∈ ℕ → {1, 𝑝} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝})
10		disjsn 4192	. . . . . . . . . . . 12 ⊢ (({1} ∩ {𝑝}) = ∅ ↔ ¬ 𝑝 ∈ {1})
11		1ex 9914	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
12	11	ensn1 7906	. . . . . . . . . . . . 13 ⊢ {1} ≈ 1𝑜
13		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑝 ∈ V
14	13	ensn1 7906	. . . . . . . . . . . . 13 ⊢ {𝑝} ≈ 1𝑜
15		pm54.43 8709	. . . . . . . . . . . . 13 ⊢ (({1} ≈ 1𝑜 ∧ {𝑝} ≈ 1𝑜) → (({1} ∩ {𝑝}) = ∅ ↔ ({1} ∪ {𝑝}) ≈ 2𝑜))
16	12, 14, 15	mp2an 704	. . . . . . . . . . . 12 ⊢ (({1} ∩ {𝑝}) = ∅ ↔ ({1} ∪ {𝑝}) ≈ 2𝑜)
17	10, 16	bitr3i 265	. . . . . . . . . . 11 ⊢ (¬ 𝑝 ∈ {1} ↔ ({1} ∪ {𝑝}) ≈ 2𝑜)
18		velsn 4141	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {1} ↔ 𝑝 = 1)
19	17, 18	xchnxbi 321	. . . . . . . . . 10 ⊢ (¬ 𝑝 = 1 ↔ ({1} ∪ {𝑝}) ≈ 2𝑜)
20		df-ne 2782	. . . . . . . . . 10 ⊢ (𝑝 ≠ 1 ↔ ¬ 𝑝 = 1)
21		df-pr 4128	. . . . . . . . . . 11 ⊢ {1, 𝑝} = ({1} ∪ {𝑝})
22	21	breq1i 4590	. . . . . . . . . 10 ⊢ ({1, 𝑝} ≈ 2𝑜 ↔ ({1} ∪ {𝑝}) ≈ 2𝑜)
23	19, 20, 22	3bitr4i 291	. . . . . . . . 9 ⊢ (𝑝 ≠ 1 ↔ {1, 𝑝} ≈ 2𝑜)
24		ensym 7891	. . . . . . . . . 10 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 → 2𝑜 ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝})
25		entr 7894	. . . . . . . . . 10 ⊢ (({1, 𝑝} ≈ 2𝑜 ∧ 2𝑜 ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}) → {1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝})
26	24, 25	sylan2 490	. . . . . . . . 9 ⊢ (({1, 𝑝} ≈ 2𝑜 ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜) → {1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝})
27	23, 26	sylanb 488	. . . . . . . 8 ⊢ ((𝑝 ≠ 1 ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜) → {1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝})
28		prfi 8120	. . . . . . . . . . 11 ⊢ {1, 𝑝} ∈ Fin
29		ensym 7891	. . . . . . . . . . 11 ⊢ ({1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ {1, 𝑝})
30		enfii 8062	. . . . . . . . . . 11 ⊢ (({1, 𝑝} ∈ Fin ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ {1, 𝑝}) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∈ Fin)
31	28, 29, 30	sylancr 694	. . . . . . . . . 10 ⊢ ({1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∈ Fin)
32	31	adantl 481	. . . . . . . . 9 ⊢ (({1, 𝑝} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∧ {1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∈ Fin)
33		dfpss2 3654	. . . . . . . . . . . 12 ⊢ ({1, 𝑝} ⊊ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ↔ ({1, 𝑝} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∧ ¬ {1, 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}))
34		pssinf 8055	. . . . . . . . . . . 12 ⊢ (({1, 𝑝} ⊊ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∧ {1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}) → ¬ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∈ Fin)
35	33, 34	sylanbr 489	. . . . . . . . . . 11 ⊢ ((({1, 𝑝} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∧ ¬ {1, 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}) ∧ {1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}) → ¬ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∈ Fin)
36	35	an32s 842	. . . . . . . . . 10 ⊢ ((({1, 𝑝} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∧ {1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}) ∧ ¬ {1, 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}) → ¬ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∈ Fin)
37	36	ex 449	. . . . . . . . 9 ⊢ (({1, 𝑝} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∧ {1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}) → (¬ {1, 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} → ¬ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∈ Fin))
38	32, 37	mt4d 151	. . . . . . . 8 ⊢ (({1, 𝑝} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ∧ {1, 𝑝} ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝}) → {1, 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝})
39	9, 27, 38	syl2an 493	. . . . . . 7 ⊢ ((𝑝 ∈ ℕ ∧ (𝑝 ≠ 1 ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜)) → {1, 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝})
40	39	eqcomd 2616	. . . . . 6 ⊢ ((𝑝 ∈ ℕ ∧ (𝑝 ≠ 1 ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜)) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝})
41	40	expr 641	. . . . 5 ⊢ ((𝑝 ∈ ℕ ∧ 𝑝 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝}))
42		breq1 4586	. . . . . . . 8 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝} → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 ↔ {1, 𝑝} ≈ 2𝑜))
43	42, 23	syl6bbr 277	. . . . . . 7 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝} → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 ↔ 𝑝 ≠ 1))
44	43	biimprcd 239	. . . . . 6 ⊢ (𝑝 ≠ 1 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝} → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜))
45	44	adantl 481	. . . . 5 ⊢ ((𝑝 ∈ ℕ ∧ 𝑝 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝} → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜))
46	41, 45	impbid 201	. . . 4 ⊢ ((𝑝 ∈ ℕ ∧ 𝑝 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝}))
47	46	ex 449	. . 3 ⊢ (𝑝 ∈ ℕ → (𝑝 ≠ 1 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {1, 𝑝})))
48	8, 47	vtoclga 3245	. 2 ⊢ (𝑃 ∈ ℕ → (𝑃 ≠ 1 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃})))
49	48	imp 444	1 ⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2𝑜 ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874  {csn 4125  {cpr 4127   class class class wbr 4583  1_𝑜c1o 7440  2_𝑜c2o 7441   ≈ cen 7838  Fincfn 7841  1c1 9816  ℕcn 10897   ∥ cdvds 14821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-z 11255  df-dvds 14822

This theorem is referenced by:  isprm2  15233