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Theorem prmdvdsfmtnof1lem2 45767
Description: Lemma 2 for prmdvdsfmtnof1 45769. (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 45716 . 2 (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
2 fmtnorn 45716 . 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 45713 . . . . . . . . . . . 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 2825 . . . . . . . . . . 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 45713 . . . . . . . . . . . 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 2825 . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . 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 2950 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ ¬ (FermatNo‘𝑛) = (FermatNo‘𝑚)) → 𝑛𝑚)
23 goldbachth 45729 . . . . . . . . . . . . . . 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 2753 . . . . . . . . . . . . . . . 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 7366 . . . . . . . . . . . . . . . 16 (((FermatNo‘𝑛) = 𝐹 ∧ (FermatNo‘𝑚) = 𝐺) → ((FermatNo‘𝑛) gcd (FermatNo‘𝑚)) = (𝐹 gcd 𝐺))
2928eqeq1d 2738 . . . . . . . . . . . . . . 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 16550 . . . . . . . . . . . 12 (𝐼 ∈ ℙ → 𝐼 ∈ ℕ)
37 coprmdvds1 16528 . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . 16 (𝐼 = 1 → (𝐼 ∈ ℙ ↔ 1 ∈ ℙ))
41 1nprm 16555 . . . . . . . . . . . . . . . . 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 3152 . . . . 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 3145 . . 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 2943  wrex 3073   class class class wbr 5105  ran crn 5634  cfv 6496  (class class class)co 7357  1c1 11052  cn 12153  0cn0 12413  cdvds 16136   gcd cgcd 16374  cprime 16547  FermatNocfmtno 45709
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-prod 15789  df-dvds 16137  df-gcd 16375  df-prm 16548  df-fmtno 45710
This theorem is referenced by:  prmdvdsfmtnof1  45769
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