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Theorem fvcosymgeq 19219
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 19165 . . . . . 6 (𝐺𝐵𝐺:𝑁𝑁)
43ffnd 6673 . . . . 5 (𝐺𝐵𝐺 Fn 𝑁)
5 gsmsymgreq.z . . . . . . 7 𝑍 = (SymGrp‘𝑀)
6 gsmsymgreq.p . . . . . . 7 𝑃 = (Base‘𝑍)
75, 6symgbasf 19165 . . . . . 6 (𝐾𝑃𝐾:𝑀𝑀)
87ffnd 6673 . . . . 5 (𝐾𝑃𝐾 Fn 𝑀)
94, 8anim12i 614 . . . 4 ((𝐺𝐵𝐾𝑃) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
109adantr 482 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
11 gsmsymgreq.i . . . . . . . 8 𝐼 = (𝑁𝑀)
1211eleq2i 2826 . . . . . . 7 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
1312biimpi 215 . . . . . 6 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
14133ad2ant1 1134 . . . . 5 ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → 𝑋 ∈ (𝑁𝑀))
1514adantl 483 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → 𝑋 ∈ (𝑁𝑀))
16 simpr2 1196 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺𝑋) = (𝐾𝑋))
171, 2symgbasf1o 19164 . . . . . . . . . . 11 (𝐺𝐵𝐺:𝑁1-1-onto𝑁)
18 dff1o5 6797 . . . . . . . . . . . 12 (𝐺:𝑁1-1-onto𝑁 ↔ (𝐺:𝑁1-1𝑁 ∧ ran 𝐺 = 𝑁))
19 eqcom 2740 . . . . . . . . . . . . 13 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2019biimpi 215 . . . . . . . . . . . 12 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2118, 20simplbiim 506 . . . . . . . . . . 11 (𝐺:𝑁1-1-onto𝑁𝑁 = ran 𝐺)
2217, 21syl 17 . . . . . . . . . 10 (𝐺𝐵𝑁 = ran 𝐺)
235, 6symgbasf1o 19164 . . . . . . . . . . 11 (𝐾𝑃𝐾:𝑀1-1-onto𝑀)
24 dff1o5 6797 . . . . . . . . . . . 12 (𝐾:𝑀1-1-onto𝑀 ↔ (𝐾:𝑀1-1𝑀 ∧ ran 𝐾 = 𝑀))
25 eqcom 2740 . . . . . . . . . . . . 13 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
2625biimpi 215 . . . . . . . . . . . 12 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
2724, 26simplbiim 506 . . . . . . . . . . 11 (𝐾:𝑀1-1-onto𝑀𝑀 = ran 𝐾)
2823, 27syl 17 . . . . . . . . . 10 (𝐾𝑃𝑀 = ran 𝐾)
2922, 28ineqan12d 4178 . . . . . . . . 9 ((𝐺𝐵𝐾𝑃) → (𝑁𝑀) = (ran 𝐺 ∩ ran 𝐾))
3011, 29eqtrid 2785 . . . . . . . 8 ((𝐺𝐵𝐾𝑃) → 𝐼 = (ran 𝐺 ∩ ran 𝐾))
3130raleqdv 3312 . . . . . . 7 ((𝐺𝐵𝐾𝑃) → (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) ↔ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
3231biimpcd 249 . . . . . 6 (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) → ((𝐺𝐵𝐾𝑃) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
33323ad2ant3 1136 . . . . 5 ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐺𝐵𝐾𝑃) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
3433impcom 409 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛))
3515, 16, 343jca 1129 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝑋 ∈ (𝑁𝑀) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
36 fvcofneq 7047 . . 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 414 1 ((𝐺𝐵𝐾𝑃) → ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  cin 3913  ran crn 5638  ccom 5641   Fn wfn 6495  1-1wf1 6497  1-1-ontowf1o 6499  cfv 6500  Basecbs 17091  SymGrpcsymg 19156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-tset 17160  df-efmnd 18687  df-symg 19157
This theorem is referenced by:  gsmsymgreqlem1  19220
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