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Theorem prmdvdsfmtnof1lem2 42073
Description: Lemma 2 for prmdvdsfmtnof1 42075. (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 42022 . 2 (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2 fmtnorn 42022 . 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 42019 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (FermatNo‘𝑛) ∈ ℕ)
65ad2antrl 710 . . . . . . . . . . 11 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (FermatNo‘𝑛) ∈ ℕ)
76adantr 468 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑛) ∈ ℕ)
8 eleq1 2880 . . . . . . . . . . 11 ((FermatNo‘𝑛) = 𝐹 → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
98ad2antll 711 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
107, 9mpbid 223 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐹 ∈ ℕ)
11 fmtnonn 42019 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0 → (FermatNo‘𝑚) ∈ ℕ)
1211ad2antll 711 . . . . . . . . . . 11 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (FermatNo‘𝑚) ∈ ℕ)
1312adantr 468 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (FermatNo‘𝑚) ∈ ℕ)
14 eleq1 2880 . . . . . . . . . . 11 ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
1514ad2antrl 710 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
1613, 15mpbid 223 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐺 ∈ ℕ)
17 simpll 774 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ∈ ℕ0)
18 simplr 776 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑚 ∈ ℕ0)
19 fveq2 6411 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (FermatNo‘𝑛) = (FermatNo‘𝑚))
2019con3i 151 . . . . . . . . . . . . . . . . 17 (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ¬ 𝑛 = 𝑚)
2120adantl 469 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ¬ 𝑛 = 𝑚)
2221neqned 2992 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛𝑚)
23 goldbachth 42035 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0𝑛𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
2417, 18, 22, 23syl3anc 1483 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
2524ex 399 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1))
26 eqeq12 2826 . . . . . . . . . . . . . . . 16 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ 𝐹 = 𝐺))
2726notbid 309 . . . . . . . . . . . . . . 15 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ ¬ 𝐹 = 𝐺))
28 oveq12 6886 . . . . . . . . . . . . . . . 16 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
2928eqeq1d 2815 . . . . . . . . . . . . . . 15 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1 ↔ (𝐹 gcd 𝐺) = 1))
3027, 29imbi12d 335 . . . . . . . . . . . . . 14 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1) ↔ (¬ 𝐹 = 𝐺 → (𝐹 gcd 𝐺) = 1)))
3130ancoms 448 . . . . . . . . . . . . 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 236 . . . . . . . . . . . 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 396 . . . . . . . . . 10 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (𝐹 gcd 𝐺) = 1))
3534imp 395 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (𝐹 gcd 𝐺) = 1)
36 prmnn 15609 . . . . . . . . . . . 12 (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37 coprmdvds1 15587 . . . . . . . . . . . . 13 ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼𝐹𝐼𝐺) → 𝐼 = 1))
3837imp 395 . . . . . . . . . . . 12 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℕ ∧ 𝐼𝐹𝐼𝐺)) → 𝐼 = 1)
3936, 38syl3anr1 1531 . . . . . . . . . . 11 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺)) → 𝐼 = 1)
40 eleq1 2880 . . . . . . . . . . . . . . . 16 (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41 1nprm 15613 . . . . . . . . . . . . . . . . 17 ¬ 1 ∈ ℙ
4241pm2.21i 117 . . . . . . . . . . . . . . . 16 (1 ∈ ℙ → 𝐹 = 𝐺)
4340, 42syl6bi 244 . . . . . . . . . . . . . . 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 1156 . . . . . . . . . . . 12 ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → (𝐼 = 1 → 𝐹 = 𝐺)))
4746impcom 396 . . . . . . . . . . 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 399 . . . . . . . . 9 ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
5010, 16, 35, 49syl3anc 1483 . . . . . . . 8 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
5150exp43 425 . . . . . . 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 3226 . . . . 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 3222 . . 3 (∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺)))
5655imp 395 . 2 ((∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 ∧ ∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
571, 2, 56syl2anb 587 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wne 2985  wrex 3104   class class class wbr 4851  ran crn 5319  cfv 6104  (class class class)co 6877  1c1 10225  cn 11308  0cn0 11562  cdvds 15206   gcd cgcd 15438  cprime 15606  FermatNocfmtno 42015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-sup 8590  df-inf 8591  df-oi 8657  df-card 9051  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-div 10973  df-nn 11309  df-2 11367  df-3 11368  df-4 11369  df-5 11370  df-n0 11563  df-z 11647  df-uz 11908  df-rp 12050  df-fz 12553  df-fzo 12693  df-seq 13028  df-exp 13087  df-hash 13341  df-cj 14065  df-re 14066  df-im 14067  df-sqrt 14201  df-abs 14202  df-clim 14445  df-prod 14860  df-dvds 15207  df-gcd 15439  df-prm 15607  df-fmtno 42016
This theorem is referenced by:  prmdvdsfmtnof1  42075
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