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Theorem prmdvdsfmtnof1lem2 46089
Description: Lemma 2 for prmdvdsfmtnof1 46091. (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 46038 . 2 (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2 fmtnorn 46038 . 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 46035 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (FermatNo‘𝑛) ∈ ℕ)
65ad2antrl 726 . . . . . . . . . . 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 2821 . . . . . . . . . . 11 ((FermatNo‘𝑛) = 𝐹 → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
98ad2antll 727 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑛) ∈ ℕ ↔ 𝐹 ∈ ℕ))
107, 9mpbid 231 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐹 ∈ ℕ)
11 fmtnonn 46035 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0 → (FermatNo‘𝑚) ∈ ℕ)
1211ad2antll 727 . . . . . . . . . . 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 2821 . . . . . . . . . . 11 ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
1514ad2antrl 726 . . . . . . . . . 10 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((FermatNo‘𝑚) ∈ ℕ ↔ 𝐺 ∈ ℕ))
1613, 15mpbid 231 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → 𝐺 ∈ ℕ)
17 simpll 765 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛 ∈ ℕ0)
18 simplr 767 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑚 ∈ ℕ0)
19 fveq2 6879 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (FermatNo‘𝑛) = (FermatNo‘𝑚))
2019con3i 154 . . . . . . . . . . . . . . . . 17 (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ¬ 𝑛 = 𝑚)
2120adantl 482 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ¬ 𝑛 = 𝑚)
2221neqned 2947 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛𝑚)
23 goldbachth 46051 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0𝑛𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
2417, 18, 22, 23syl3anc 1371 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1)
2524ex 413 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1))
26 eqeq12 2749 . . . . . . . . . . . . . . . 16 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ 𝐹 = 𝐺))
2726notbid 317 . . . . . . . . . . . . . . 15 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) ↔ ¬ 𝐹 = 𝐺))
28 oveq12 7403 . . . . . . . . . . . . . . . 16 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
2928eqeq1d 2734 . . . . . . . . . . . . . . 15 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → (((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1 ↔ (𝐹 gcd 𝐺) = 1))
3027, 29imbi12d 344 . . . . . . . . . . . . . 14 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((¬ (FermatNo‘𝑛) = (FermatNo‘𝑚) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = 1) ↔ (¬ 𝐹 = 𝐺 → (𝐹 gcd 𝐺) = 1)))
3130ancoms 459 . . . . . . . . . . . . 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 244 . . . . . . . . . . . 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 408 . . . . . . . . . 10 ((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → (((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹) → (𝐹 gcd 𝐺) = 1))
3534imp 407 . . . . . . . . 9 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → (𝐹 gcd 𝐺) = 1)
36 prmnn 16595 . . . . . . . . . . . 12 (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37 coprmdvds1 16573 . . . . . . . . . . . . 13 ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼𝐹𝐼𝐺) → 𝐼 = 1))
3837imp 407 . . . . . . . . . . . 12 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℕ ∧ 𝐼𝐹𝐼𝐺)) → 𝐼 = 1)
3936, 38syl3anr1 1416 . . . . . . . . . . 11 (((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) ∧ (𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺)) → 𝐼 = 1)
40 eleq1 2821 . . . . . . . . . . . . . . . 16 (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41 1nprm 16600 . . . . . . . . . . . . . . . . 17 ¬ 1 ∈ ℙ
4241pm2.21i 119 . . . . . . . . . . . . . . . 16 (1 ∈ ℙ → 𝐹 = 𝐺)
4340, 42syl6bi 252 . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . 12 ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → (𝐼 = 1 → 𝐹 = 𝐺)))
4746impcom 408 . . . . . . . . . . 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 413 . . . . . . . . 9 ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
5010, 16, 35, 49syl3anc 1371 . . . . . . . 8 (((¬ 𝐹 = 𝐺 ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) ∧ ((FermatNo‘𝑚) = 𝐺 ∧ (FermatNo‘𝑛) = 𝐹)) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
5150exp43 437 . . . . . . 7 𝐹 = 𝐺 → ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺)))))
524, 51pm2.61i 182 . . . . . 6 ((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) → ((FermatNo‘𝑚) = 𝐺 → ((FermatNo‘𝑛) = 𝐹 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))))
5352rexlimdva 3155 . . . . 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 3148 . . 3 (∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 → (∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺 → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺)))
5655imp 407 . 2 ((∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹 ∧ ∃𝑚 ∈ ℕ0 (FermatNo‘𝑚) = 𝐺) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))
571, 2, 56syl2anb 598 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wrex 3070   class class class wbr 5142  ran crn 5671  cfv 6533  (class class class)co 7394  1c1 11095  cn 12196  0cn0 12456  cdvds 16181   gcd cgcd 16419  cprime 16592  FermatNocfmtno 46031
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-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-inf2 9620  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171  ax-pre-sup 11172
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 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-2o 8451  df-er 8688  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-sup 9421  df-inf 9422  df-oi 9489  df-card 9918  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-div 11856  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-n0 12457  df-z 12543  df-uz 12807  df-rp 12959  df-fz 13469  df-fzo 13612  df-seq 13951  df-exp 14012  df-hash 14275  df-cj 15030  df-re 15031  df-im 15032  df-sqrt 15166  df-abs 15167  df-clim 15416  df-prod 15834  df-dvds 16182  df-gcd 16420  df-prm 16593  df-fmtno 46032
This theorem is referenced by:  prmdvdsfmtnof1  46091
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