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Theorem prmdvdsfmtnof1lem2 40784
Description: Lemma 2 for prmdvdsfmtnof1 40786. (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 40733 . 2 (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2 fmtnorn 40733 . 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 40730 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (FermatNo‘𝑛) ∈ ℕ)
65ad2antrl 763 . . . . . . . . . . 11 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (FermatNo‘𝑛) ∈ ℕ)
76adantr 481 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑛) ∈ ℕ)
8 eleq1 2692 . . . . . . . . . . 11 ((FermatNo‘𝑛) = 𝐹 → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
98ad2antll 764 . . . . . . . . . 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 40730 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0 → (FermatNo‘𝑚) ∈ ℕ)
1211ad2antll 764 . . . . . . . . . . 11 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (FermatNo‘𝑚) ∈ ℕ)
1312adantr 481 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑚) ∈ ℕ)
14 eleq1 2692 . . . . . . . . . . 11 ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
1514ad2antrl 763 . . . . . . . . . 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 789 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ∈ ℕ0)
18 simplr 791 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑚 ∈ ℕ0)
19 fveq2 6150 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (FermatNo‘𝑛) = (FermatNo‘𝑚))
2019con3i 150 . . . . . . . . . . . . . . . . 17 (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ¬ 𝑛 = 𝑚)
2120adantl 482 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ¬ 𝑛 = 𝑚)
2221neqned 2803 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛𝑚)
23 goldbachth 40746 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0𝑛𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
2417, 18, 22, 23syl3anc 1323 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
2524ex 450 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1))
26 eqeq12 2639 . . . . . . . . . . . . . . . 16 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ 𝐹 = 𝐺))
2726notbid 308 . . . . . . . . . . . . . . 15 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ ¬ 𝐹 = 𝐺))
28 oveq12 6614 . . . . . . . . . . . . . . . 16 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
2928eqeq1d 2628 . . . . . . . . . . . . . . 15 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1 ↔ (𝐹 gcd 𝐺) = 1))
3027, 29imbi12d 334 . . . . . . . . . . . . . 14 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1) ↔ (¬ 𝐹 = 𝐺 → (𝐹 gcd 𝐺) = 1)))
3130ancoms 469 . . . . . . . . . . . . 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 446 . . . . . . . . . 10 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (𝐹 gcd 𝐺) = 1))
3534imp 445 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (𝐹 gcd 𝐺) = 1)
36 prmnn 15307 . . . . . . . . . . . 12 (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37 coprmdvds1 15284 . . . . . . . . . . . . 13 ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼𝐹𝐼𝐺) → 𝐼 = 1))
3837imp 445 . . . . . . . . . . . 12 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℕ ∧ 𝐼𝐹𝐼𝐺)) → 𝐼 = 1)
3936, 38syl3anr1 1375 . . . . . . . . . . 11 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺)) → 𝐼 = 1)
40 eleq1 2692 . . . . . . . . . . . . . . . 16 (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41 1nprm 15311 . . . . . . . . . . . . . . . . 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 1080 . . . . . . . . . . . 12 ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → (𝐼 = 1 → 𝐹 = 𝐺)))
4746impcom 446 . . . . . . . . . . 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 450 . . . . . . . . 9 ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
5010, 16, 35, 49syl3anc 1323 . . . . . . . 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 3029 . . . . 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 3025 . . 3 (∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺)))
5655imp 445 . 2 ((∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 ∧ ∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
571, 2, 56syl2anb 496 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 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wrex 2913   class class class wbr 4618  ran crn 5080  cfv 5850  (class class class)co 6605  1c1 9882  cn 10965  0cn0 11237  cdvds 14902   gcd cgcd 15135  cprime 15304  FermatNocfmtno 40726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-inf 8294  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-clim 14148  df-prod 14556  df-dvds 14903  df-gcd 15136  df-prm 15305  df-fmtno 40727
This theorem is referenced by:  prmdvdsfmtnof1  40786
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