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Theorem fvcosymgeq 17672
 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 17627 . . . . . 6 (𝐺𝐵𝐺:𝑁𝑁)
4 ffn 5958 . . . . . 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 17627 . . . . . 6 (𝐾𝑃𝐾:𝑀𝑀)
9 ffn 5958 . . . . . 6 (𝐾:𝑀𝑀𝐾 Fn 𝑀)
108, 9syl 17 . . . . 5 (𝐾𝑃𝐾 Fn 𝑀)
115, 10anim12i 588 . . . 4 ((𝐺𝐵𝐾𝑃) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
1211adantr 480 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺 Fn 𝑁𝐾 Fn 𝑀))
13 gsmsymgreq.i . . . . . . . 8 𝐼 = (𝑁𝑀)
1413eleq2i 2680 . . . . . . 7 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
1514biimpi 205 . . . . . 6 (𝑋𝐼𝑋 ∈ (𝑁𝑀))
16153ad2ant1 1075 . . . . 5 ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → 𝑋 ∈ (𝑁𝑀))
1716adantl 481 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → 𝑋 ∈ (𝑁𝑀))
18 simpr2 1061 . . . 4 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝐺𝑋) = (𝐾𝑋))
191, 2symgbasf1o 17626 . . . . . . . . . . 11 (𝐺𝐵𝐺:𝑁1-1-onto𝑁)
20 dff1o5 6059 . . . . . . . . . . . 12 (𝐺:𝑁1-1-onto𝑁 ↔ (𝐺:𝑁1-1𝑁 ∧ ran 𝐺 = 𝑁))
21 eqcom 2617 . . . . . . . . . . . . . 14 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2221biimpi 205 . . . . . . . . . . . . 13 (ran 𝐺 = 𝑁𝑁 = ran 𝐺)
2322adantl 481 . . . . . . . . . . . 12 ((𝐺:𝑁1-1𝑁 ∧ ran 𝐺 = 𝑁) → 𝑁 = ran 𝐺)
2420, 23sylbi 206 . . . . . . . . . . 11 (𝐺:𝑁1-1-onto𝑁𝑁 = ran 𝐺)
2519, 24syl 17 . . . . . . . . . 10 (𝐺𝐵𝑁 = ran 𝐺)
266, 7symgbasf1o 17626 . . . . . . . . . . 11 (𝐾𝑃𝐾:𝑀1-1-onto𝑀)
27 dff1o5 6059 . . . . . . . . . . . 12 (𝐾:𝑀1-1-onto𝑀 ↔ (𝐾:𝑀1-1𝑀 ∧ ran 𝐾 = 𝑀))
28 eqcom 2617 . . . . . . . . . . . . . 14 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
2928biimpi 205 . . . . . . . . . . . . 13 (ran 𝐾 = 𝑀𝑀 = ran 𝐾)
3029adantl 481 . . . . . . . . . . . 12 ((𝐾:𝑀1-1𝑀 ∧ ran 𝐾 = 𝑀) → 𝑀 = ran 𝐾)
3127, 30sylbi 206 . . . . . . . . . . 11 (𝐾:𝑀1-1-onto𝑀𝑀 = ran 𝐾)
3226, 31syl 17 . . . . . . . . . 10 (𝐾𝑃𝑀 = ran 𝐾)
3325, 32ineqan12d 3778 . . . . . . . . 9 ((𝐺𝐵𝐾𝑃) → (𝑁𝑀) = (ran 𝐺 ∩ ran 𝐾))
3413, 33syl5eq 2656 . . . . . . . 8 ((𝐺𝐵𝐾𝑃) → 𝐼 = (ran 𝐺 ∩ ran 𝐾))
3534raleqdv 3121 . . . . . . 7 ((𝐺𝐵𝐾𝑃) → (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) ↔ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
3635biimpcd 238 . . . . . 6 (∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛) → ((𝐺𝐵𝐾𝑃) → ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
37363ad2ant3 1077 . . . . 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 1235 . . 3 (((𝐺𝐵𝐾𝑃) ∧ (𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛))) → (𝑋 ∈ (𝑁𝑀) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)))
40 fvcofneq 6275 . . 3 ((𝐺 Fn 𝑁𝐾 Fn 𝑀) → ((𝑋 ∈ (𝑁𝑀) ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
4112, 39, 40sylc 63 . 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 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  Basecbs 15695  SymGrpcsymg 17620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-tset 15787  df-symg 17621 This theorem is referenced by:  gsmsymgreqlem1  17673
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