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Theorem fvcosymgeq 18056
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 18011 . . . . . 6 (𝐺𝐵𝐺:𝑁𝑁)
43ffnd 6185 . . . . 5 (𝐺𝐵𝐺 Fn 𝑁)
5 gsmsymgreq.z . . . . . . 7 𝑍 = (SymGrp‘𝑀)
6 gsmsymgreq.p . . . . . . 7 𝑃 = (Base‘𝑍)
75, 6symgbasf 18011 . . . . . 6 (𝐾𝑃𝐾:𝑀𝑀)
87ffnd 6185 . . . . 5 (𝐾𝑃𝐾 Fn 𝑀)
94, 8anim12i 600 . . . 4 ((𝐺𝐵𝐾𝑃) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
109adantr 466 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
11 gsmsymgreq.i . . . . . . . 8 𝐼 = (𝑁𝑀)
1211eleq2i 2842 . . . . . . 7 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
1312biimpi 206 . . . . . 6 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
14133ad2ant1 1127 . . . . 5 ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → 𝑋 ∈ (𝑁𝑀))
1514adantl 467 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → 𝑋 ∈ (𝑁𝑀))
16 simpr2 1235 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺𝑋) = (𝐾𝑋))
171, 2symgbasf1o 18010 . . . . . . . . . . 11 (𝐺𝐵𝐺:𝑁1-1-onto𝑁)
18 dff1o5 6288 . . . . . . . . . . . 12 (𝐺:𝑁1-1-onto𝑁 ↔ (𝐺:𝑁1-1𝑁 ∧ ran 𝐺 = 𝑁))
19 eqcom 2778 . . . . . . . . . . . . . 14 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2019biimpi 206 . . . . . . . . . . . . 13 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2120adantl 467 . . . . . . . . . . . 12 ((𝐺:𝑁1-1𝑁 ∧ ran 𝐺 = 𝑁) → 𝑁 = ran 𝐺)
2218, 21sylbi 207 . . . . . . . . . . 11 (𝐺:𝑁1-1-onto𝑁𝑁 = ran 𝐺)
2317, 22syl 17 . . . . . . . . . 10 (𝐺𝐵𝑁 = ran 𝐺)
245, 6symgbasf1o 18010 . . . . . . . . . . 11 (𝐾𝑃𝐾:𝑀1-1-onto𝑀)
25 dff1o5 6288 . . . . . . . . . . . 12 (𝐾:𝑀1-1-onto𝑀 ↔ (𝐾:𝑀1-1𝑀 ∧ ran 𝐾 = 𝑀))
26 eqcom 2778 . . . . . . . . . . . . . 14 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
2726biimpi 206 . . . . . . . . . . . . 13 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
2827adantl 467 . . . . . . . . . . . 12 ((𝐾:𝑀1-1𝑀 ∧ ran 𝐾 = 𝑀) → 𝑀 = ran 𝐾)
2925, 28sylbi 207 . . . . . . . . . . 11 (𝐾:𝑀1-1-onto𝑀𝑀 = ran 𝐾)
3024, 29syl 17 . . . . . . . . . 10 (𝐾𝑃𝑀 = ran 𝐾)
3123, 30ineqan12d 3967 . . . . . . . . 9 ((𝐺𝐵𝐾𝑃) → (𝑁𝑀) = (ran 𝐺 ∩ ran 𝐾))
3211, 31syl5eq 2817 . . . . . . . 8 ((𝐺𝐵𝐾𝑃) → 𝐼 = (ran 𝐺 ∩ ran 𝐾))
3332raleqdv 3293 . . . . . . 7 ((𝐺𝐵𝐾𝑃) → (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) ↔ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
3433biimpcd 239 . . . . . 6 (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) → ((𝐺𝐵𝐾𝑃) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
35343ad2ant3 1129 . . . . 5 ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐺𝐵𝐾𝑃) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
3635impcom 394 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛))
3715, 16, 363jca 1122 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝑋 ∈ (𝑁𝑀) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
38 fvcofneq 6512 . . 3 ((𝐺 Fn 𝑁𝐾 Fn 𝑀) → ((𝑋 ∈ (𝑁𝑀) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
3910, 37, 38sylc 65 . 2 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋))
4039ex 397 1 ((𝐺𝐵𝐾𝑃) → ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  cin 3722  ran crn 5251  ccom 5254   Fn wfn 6025  1-1wf1 6027  1-1-ontowf1o 6029  cfv 6030  Basecbs 16064  SymGrpcsymg 18004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-7 11290  df-8 11291  df-9 11292  df-n0 11500  df-z 11585  df-uz 11894  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-plusg 16162  df-tset 16168  df-symg 18005
This theorem is referenced by:  gsmsymgreqlem1  18057
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