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Theorem fvcosymgeq 18047
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 18002 . . . . . 6 (𝐺𝐵𝐺:𝑁𝑁)
4 ffn 6204 . . . . . 6 (𝐺:𝑁𝑁𝐺 Fn 𝑁)
53, 4syl 17 . . . . 5 (𝐺𝐵𝐺 Fn 𝑁)
6 gsmsymgreq.z . . . . . . 7 𝑍 = (SymGrp‘𝑀)
7 gsmsymgreq.p . . . . . . 7 𝑃 = (Base‘𝑍)
86, 7symgbasf 18002 . . . . . 6 (𝐾𝑃𝐾:𝑀𝑀)
9 ffn 6204 . . . . . 6 (𝐾:𝑀𝑀𝐾 Fn 𝑀)
108, 9syl 17 . . . . 5 (𝐾𝑃𝐾 Fn 𝑀)
115, 10anim12i 591 . . . 4 ((𝐺𝐵𝐾𝑃) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
1211adantr 472 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
13 gsmsymgreq.i . . . . . . . 8 𝐼 = (𝑁𝑀)
1413eleq2i 2829 . . . . . . 7 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
1514biimpi 206 . . . . . 6 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
16153ad2ant1 1128 . . . . 5 ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → 𝑋 ∈ (𝑁𝑀))
1716adantl 473 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → 𝑋 ∈ (𝑁𝑀))
18 simpr2 1236 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺𝑋) = (𝐾𝑋))
191, 2symgbasf1o 18001 . . . . . . . . . . 11 (𝐺𝐵𝐺:𝑁1-1-onto𝑁)
20 dff1o5 6305 . . . . . . . . . . . 12 (𝐺:𝑁1-1-onto𝑁 ↔ (𝐺:𝑁1-1𝑁 ∧ ran 𝐺 = 𝑁))
21 eqcom 2765 . . . . . . . . . . . . . 14 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2221biimpi 206 . . . . . . . . . . . . 13 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2322adantl 473 . . . . . . . . . . . 12 ((𝐺:𝑁1-1𝑁 ∧ ran 𝐺 = 𝑁) → 𝑁 = ran 𝐺)
2420, 23sylbi 207 . . . . . . . . . . 11 (𝐺:𝑁1-1-onto𝑁𝑁 = ran 𝐺)
2519, 24syl 17 . . . . . . . . . 10 (𝐺𝐵𝑁 = ran 𝐺)
266, 7symgbasf1o 18001 . . . . . . . . . . 11 (𝐾𝑃𝐾:𝑀1-1-onto𝑀)
27 dff1o5 6305 . . . . . . . . . . . 12 (𝐾:𝑀1-1-onto𝑀 ↔ (𝐾:𝑀1-1𝑀 ∧ ran 𝐾 = 𝑀))
28 eqcom 2765 . . . . . . . . . . . . . 14 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
2928biimpi 206 . . . . . . . . . . . . 13 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
3029adantl 473 . . . . . . . . . . . 12 ((𝐾:𝑀1-1𝑀 ∧ ran 𝐾 = 𝑀) → 𝑀 = ran 𝐾)
3127, 30sylbi 207 . . . . . . . . . . 11 (𝐾:𝑀1-1-onto𝑀𝑀 = ran 𝐾)
3226, 31syl 17 . . . . . . . . . 10 (𝐾𝑃𝑀 = ran 𝐾)
3325, 32ineqan12d 3957 . . . . . . . . 9 ((𝐺𝐵𝐾𝑃) → (𝑁𝑀) = (ran 𝐺 ∩ ran 𝐾))
3413, 33syl5eq 2804 . . . . . . . 8 ((𝐺𝐵𝐾𝑃) → 𝐼 = (ran 𝐺 ∩ ran 𝐾))
3534raleqdv 3281 . . . . . . 7 ((𝐺𝐵𝐾𝑃) → (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) ↔ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
3635biimpcd 239 . . . . . 6 (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) → ((𝐺𝐵𝐾𝑃) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
37363ad2ant3 1130 . . . . 5 ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐺𝐵𝐾𝑃) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
3837impcom 445 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛))
3917, 18, 383jca 1123 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝑋 ∈ (𝑁𝑀) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
40 fvcofneq 6528 . . 3 ((𝐺 Fn 𝑁𝐾 Fn 𝑀) → ((𝑋 ∈ (𝑁𝑀) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
4112, 39, 40sylc 65 . 2 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋))
4241ex 449 1 ((𝐺𝐵𝐾𝑃) → ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048  cin 3712  ran crn 5265  ccom 5268   Fn wfn 6042  wf 6043  1-1wf1 6044  1-1-ontowf1o 6046  cfv 6047  Basecbs 16057  SymGrpcsymg 17995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-oadd 7731  df-er 7909  df-map 8023  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-z 11568  df-uz 11878  df-fz 12518  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-plusg 16154  df-tset 16160  df-symg 17996
This theorem is referenced by:  gsmsymgreqlem1  18048
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