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Theorem prmdvdsfmtnof1lem2 41822
Description: Lemma 2 for prmdvdsfmtnof1 41824. (Contributed by AV, 3-Aug-2021.)
Assertion
Ref Expression
prmdvdsfmtnof1lem2 ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))

Proof of Theorem prmdvdsfmtnof1lem2
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmtnorn 41771 . 2 (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2 fmtnorn 41771 . 2 (𝐺 ∈ ran FermatNo ↔ ∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺)
3 2a1 28 . . . . . . . 8 (𝐹 = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺)))
432a1d 26 . . . . . . 7 (𝐹 = 𝐺 → ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺)))))
5 fmtnonn 41768 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (FermatNo‘𝑛) ∈ ℕ)
65ad2antrl 764 . . . . . . . . . . 11 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (FermatNo‘𝑛) ∈ ℕ)
76adantr 480 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑛) ∈ ℕ)
8 eleq1 2718 . . . . . . . . . . 11 ((FermatNo‘𝑛) = 𝐹 → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
98ad2antll 765 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
107, 9mpbid 222 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐹 ∈ ℕ)
11 fmtnonn 41768 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0 → (FermatNo‘𝑚) ∈ ℕ)
1211ad2antll 765 . . . . . . . . . . 11 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (FermatNo‘𝑚) ∈ ℕ)
1312adantr 480 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑚) ∈ ℕ)
14 eleq1 2718 . . . . . . . . . . 11 ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
1514ad2antrl 764 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
1613, 15mpbid 222 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐺 ∈ ℕ)
17 simpll 805 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ∈ ℕ0)
18 simplr 807 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑚 ∈ ℕ0)
19 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (FermatNo‘𝑛) = (FermatNo‘𝑚))
2019con3i 150 . . . . . . . . . . . . . . . . 17 (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ¬ 𝑛 = 𝑚)
2120adantl 481 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ¬ 𝑛 = 𝑚)
2221neqned 2830 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛𝑚)
23 goldbachth 41784 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0𝑛𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
2417, 18, 22, 23syl3anc 1366 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
2524ex 449 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1))
26 eqeq12 2664 . . . . . . . . . . . . . . . 16 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ 𝐹 = 𝐺))
2726notbid 307 . . . . . . . . . . . . . . 15 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ ¬ 𝐹 = 𝐺))
28 oveq12 6699 . . . . . . . . . . . . . . . 16 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
2928eqeq1d 2653 . . . . . . . . . . . . . . 15 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1 ↔ (𝐹 gcd 𝐺) = 1))
3027, 29imbi12d 333 . . . . . . . . . . . . . 14 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1) ↔ (¬ 𝐹 = 𝐺 → (𝐹 gcd 𝐺) = 1)))
3130ancoms 468 . . . . . . . . . . . . 13 (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → ((¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1) ↔ (¬ 𝐹 = 𝐺 → (𝐹 gcd 𝐺) = 1)))
3225, 31syl5ibcom 235 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (¬ 𝐹 = 𝐺 → (𝐹 gcd 𝐺) = 1)))
3332com23 86 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → (¬ 𝐹 = 𝐺 → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (𝐹 gcd 𝐺) = 1)))
3433impcom 445 . . . . . . . . . 10 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (𝐹 gcd 𝐺) = 1))
3534imp 444 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (𝐹 gcd 𝐺) = 1)
36 prmnn 15435 . . . . . . . . . . . 12 (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37 coprmdvds1 15412 . . . . . . . . . . . . 13 ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼𝐹𝐼𝐺) → 𝐼 = 1))
3837imp 444 . . . . . . . . . . . 12 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℕ ∧ 𝐼𝐹𝐼𝐺)) → 𝐼 = 1)
3936, 38syl3anr1 1418 . . . . . . . . . . 11 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺)) → 𝐼 = 1)
40 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41 1nprm 15439 . . . . . . . . . . . . . . . . 17 ¬ 1 ∈ ℙ
4241pm2.21i 116 . . . . . . . . . . . . . . . 16 (1 ∈ ℙ → 𝐹 = 𝐺)
4340, 42syl6bi 243 . . . . . . . . . . . . . . 15 (𝐼 = 1 → (𝐼 ∈ ℙ → 𝐹 = 𝐺))
4443com12 32 . . . . . . . . . . . . . 14 (𝐼 ∈ ℙ → (𝐼 = 1 → 𝐹 = 𝐺))
4544a1d 25 . . . . . . . . . . . . 13 (𝐼 ∈ ℙ → ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → (𝐼 = 1 → 𝐹 = 𝐺)))
46453ad2ant1 1102 . . . . . . . . . . . 12 ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → (𝐼 = 1 → 𝐹 = 𝐺)))
4746impcom 445 . . . . . . . . . . 11 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺)) → (𝐼 = 1 → 𝐹 = 𝐺))
4839, 47mpd 15 . . . . . . . . . 10 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺)) → 𝐹 = 𝐺)
4948ex 449 . . . . . . . . 9 ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
5010, 16, 35, 49syl3anc 1366 . . . . . . . 8 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
5150exp43 639 . . . . . . 7 𝐹 = 𝐺 → ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺)))))
524, 51pm2.61i 176 . . . . . 6 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))))
5352rexlimdva 3060 . . . . 5 (𝑛 ∈ ℕ0 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))))
5453com23 86 . . . 4 (𝑛 ∈ ℕ0 → ((FermatNo‘𝑛) = 𝐹 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))))
5554rexlimiv 3056 . . 3 (∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺)))
5655imp 444 . 2 ((∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 ∧ ∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
571, 2, 56syl2anb 495 1 ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942   class class class wbr 4685  ran crn 5144  cfv 5926  (class class class)co 6690  1c1 9975  cn 11058  0cn0 11330  cdvds 15027   gcd cgcd 15263  cprime 15432  FermatNocfmtno 41764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-prod 14680  df-dvds 15028  df-gcd 15264  df-prm 15433  df-fmtno 41765
This theorem is referenced by:  prmdvdsfmtnof1  41824
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