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Theorem fvcosymgeq 19202
Description: The values of two compositions of permutations are equal if the values of the composed permutations are pairwise equal. (Contributed by AV, 26-Jan-2019.)
Hypotheses
Ref Expression
gsmsymgrfix.s 𝑆 = (SymGrp‘𝑁)
gsmsymgrfix.b 𝐵 = (Base‘𝑆)
gsmsymgreq.z 𝑍 = (SymGrp‘𝑀)
gsmsymgreq.p 𝑃 = (Base‘𝑍)
gsmsymgreq.i 𝐼 = (𝑁𝑀)
Assertion
Ref Expression
fvcosymgeq ((𝐺𝐵𝐾𝑃) → ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
Distinct variable groups:   𝑛,𝐹   𝑛,𝐺   𝑛,𝐻   𝑛,𝐼   𝑛,𝐾   𝑛,𝑋
Allowed substitution hints:   𝐵(𝑛)   𝑃(𝑛)   𝑆(𝑛)   𝑀(𝑛)   𝑁(𝑛)   𝑍(𝑛)

Proof of Theorem fvcosymgeq
StepHypRef Expression
1 gsmsymgrfix.s . . . . . . 7 𝑆 = (SymGrp‘𝑁)
2 gsmsymgrfix.b . . . . . . 7 𝐵 = (Base‘𝑆)
31, 2symgbasf 19148 . . . . . 6 (𝐺𝐵𝐺:𝑁𝑁)
43ffnd 6666 . . . . 5 (𝐺𝐵𝐺 Fn 𝑁)
5 gsmsymgreq.z . . . . . . 7 𝑍 = (SymGrp‘𝑀)
6 gsmsymgreq.p . . . . . . 7 𝑃 = (Base‘𝑍)
75, 6symgbasf 19148 . . . . . 6 (𝐾𝑃𝐾:𝑀𝑀)
87ffnd 6666 . . . . 5 (𝐾𝑃𝐾 Fn 𝑀)
94, 8anim12i 613 . . . 4 ((𝐺𝐵𝐾𝑃) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
109adantr 481 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
11 gsmsymgreq.i . . . . . . . 8 𝐼 = (𝑁𝑀)
1211eleq2i 2829 . . . . . . 7 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
1312biimpi 215 . . . . . 6 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
14133ad2ant1 1133 . . . . 5 ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → 𝑋 ∈ (𝑁𝑀))
1514adantl 482 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → 𝑋 ∈ (𝑁𝑀))
16 simpr2 1195 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺𝑋) = (𝐾𝑋))
171, 2symgbasf1o 19147 . . . . . . . . . . 11 (𝐺𝐵𝐺:𝑁1-1-onto𝑁)
18 dff1o5 6790 . . . . . . . . . . . 12 (𝐺:𝑁1-1-onto𝑁 ↔ (𝐺:𝑁1-1𝑁 ∧ ran 𝐺 = 𝑁))
19 eqcom 2743 . . . . . . . . . . . . 13 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2019biimpi 215 . . . . . . . . . . . 12 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2118, 20simplbiim 505 . . . . . . . . . . 11 (𝐺:𝑁1-1-onto𝑁𝑁 = ran 𝐺)
2217, 21syl 17 . . . . . . . . . 10 (𝐺𝐵𝑁 = ran 𝐺)
235, 6symgbasf1o 19147 . . . . . . . . . . 11 (𝐾𝑃𝐾:𝑀1-1-onto𝑀)
24 dff1o5 6790 . . . . . . . . . . . 12 (𝐾:𝑀1-1-onto𝑀 ↔ (𝐾:𝑀1-1𝑀 ∧ ran 𝐾 = 𝑀))
25 eqcom 2743 . . . . . . . . . . . . 13 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
2625biimpi 215 . . . . . . . . . . . 12 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
2724, 26simplbiim 505 . . . . . . . . . . 11 (𝐾:𝑀1-1-onto𝑀𝑀 = ran 𝐾)
2823, 27syl 17 . . . . . . . . . 10 (𝐾𝑃𝑀 = ran 𝐾)
2922, 28ineqan12d 4172 . . . . . . . . 9 ((𝐺𝐵𝐾𝑃) → (𝑁𝑀) = (ran 𝐺 ∩ ran 𝐾))
3011, 29eqtrid 2788 . . . . . . . 8 ((𝐺𝐵𝐾𝑃) → 𝐼 = (ran 𝐺 ∩ ran 𝐾))
3130raleqdv 3311 . . . . . . 7 ((𝐺𝐵𝐾𝑃) → (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) ↔ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
3231biimpcd 248 . . . . . 6 (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) → ((𝐺𝐵𝐾𝑃) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
33323ad2ant3 1135 . . . . 5 ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐺𝐵𝐾𝑃) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
3433impcom 408 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛))
3515, 16, 343jca 1128 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝑋 ∈ (𝑁𝑀) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
36 fvcofneq 7039 . . 3 ((𝐺 Fn 𝑁𝐾 Fn 𝑀) → ((𝑋 ∈ (𝑁𝑀) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
3710, 35, 36sylc 65 . 2 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋))
3837ex 413 1 ((𝐺𝐵𝐾𝑃) → ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3062  cin 3907  ran crn 5632  ccom 5635   Fn wfn 6488  1-1wf1 6490  1-1-ontowf1o 6492  cfv 6493  Basecbs 17075  SymGrpcsymg 19139
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-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124
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 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-1o 8408  df-er 8644  df-map 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12410  df-z 12496  df-uz 12760  df-fz 13417  df-struct 17011  df-sets 17028  df-slot 17046  df-ndx 17058  df-base 17076  df-ress 17105  df-plusg 17138  df-tset 17144  df-efmnd 18671  df-symg 19140
This theorem is referenced by:  gsmsymgreqlem1  19203
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