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Theorem frgpcpbl 17995
 Description: Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
frgpval.m 𝐺 = (freeGrp‘𝐼)
frgpval.b 𝑀 = (freeMnd‘(𝐼 × 2𝑜))
frgpval.r = ( ~FG𝐼)
frgpcpbl.p + = (+g𝑀)
Assertion
Ref Expression
frgpcpbl ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷))

Proof of Theorem frgpcpbl
Dummy variables 𝑘 𝑚 𝑛 𝑡 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 ( I ‘Word (𝐼 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜))
2 frgpval.r . . 3 = ( ~FG𝐼)
3 eqid 2610 . . 3 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
4 eqid 2610 . . 3 (𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩))) = (𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩)))
5 eqid 2610 . . 3 (( I ‘Word (𝐼 × 2𝑜)) ∖ 𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥)) = (( I ‘Word (𝐼 × 2𝑜)) ∖ 𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥))
6 eqid 2610 . . 3 (𝑚 ∈ {𝑡 ∈ (Word ( I ‘Word (𝐼 × 2𝑜)) ∖ {∅}) ∣ ((𝑡‘0) ∈ (( I ‘Word (𝐼 × 2𝑜)) ∖ 𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥)) ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩)))‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1))) = (𝑚 ∈ {𝑡 ∈ (Word ( I ‘Word (𝐼 × 2𝑜)) ∖ {∅}) ∣ ((𝑡‘0) ∈ (( I ‘Word (𝐼 × 2𝑜)) ∖ 𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥)) ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩)))‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
71, 2, 3, 4, 5, 6efgcpbl2 17993 . 2 ((𝐴 𝐶𝐵 𝐷) → (𝐴 ++ 𝐵) (𝐶 ++ 𝐷))
81, 2efger 17954 . . . . . 6 Er ( I ‘Word (𝐼 × 2𝑜))
98a1i 11 . . . . 5 ((𝐴 𝐶𝐵 𝐷) → Er ( I ‘Word (𝐼 × 2𝑜)))
10 simpl 472 . . . . 5 ((𝐴 𝐶𝐵 𝐷) → 𝐴 𝐶)
119, 10ercl 7640 . . . 4 ((𝐴 𝐶𝐵 𝐷) → 𝐴 ∈ ( I ‘Word (𝐼 × 2𝑜)))
121efgrcl 17951 . . . . . . 7 (𝐴 ∈ ( I ‘Word (𝐼 × 2𝑜)) → (𝐼 ∈ V ∧ ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜)))
1311, 12syl 17 . . . . . 6 ((𝐴 𝐶𝐵 𝐷) → (𝐼 ∈ V ∧ ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜)))
1413simprd 478 . . . . 5 ((𝐴 𝐶𝐵 𝐷) → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜))
1513simpld 474 . . . . . . 7 ((𝐴 𝐶𝐵 𝐷) → 𝐼 ∈ V)
16 2on 7455 . . . . . . 7 2𝑜 ∈ On
17 xpexg 6858 . . . . . . 7 ((𝐼 ∈ V ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
1815, 16, 17sylancl 693 . . . . . 6 ((𝐴 𝐶𝐵 𝐷) → (𝐼 × 2𝑜) ∈ V)
19 frgpval.b . . . . . . 7 𝑀 = (freeMnd‘(𝐼 × 2𝑜))
20 eqid 2610 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
2119, 20frmdbas 17212 . . . . . 6 ((𝐼 × 2𝑜) ∈ V → (Base‘𝑀) = Word (𝐼 × 2𝑜))
2218, 21syl 17 . . . . 5 ((𝐴 𝐶𝐵 𝐷) → (Base‘𝑀) = Word (𝐼 × 2𝑜))
2314, 22eqtr4d 2647 . . . 4 ((𝐴 𝐶𝐵 𝐷) → ( I ‘Word (𝐼 × 2𝑜)) = (Base‘𝑀))
2411, 23eleqtrd 2690 . . 3 ((𝐴 𝐶𝐵 𝐷) → 𝐴 ∈ (Base‘𝑀))
25 simpr 476 . . . . 5 ((𝐴 𝐶𝐵 𝐷) → 𝐵 𝐷)
269, 25ercl 7640 . . . 4 ((𝐴 𝐶𝐵 𝐷) → 𝐵 ∈ ( I ‘Word (𝐼 × 2𝑜)))
2726, 23eleqtrd 2690 . . 3 ((𝐴 𝐶𝐵 𝐷) → 𝐵 ∈ (Base‘𝑀))
28 frgpcpbl.p . . . 4 + = (+g𝑀)
2919, 20, 28frmdadd 17215 . . 3 ((𝐴 ∈ (Base‘𝑀) ∧ 𝐵 ∈ (Base‘𝑀)) → (𝐴 + 𝐵) = (𝐴 ++ 𝐵))
3024, 27, 29syl2anc 691 . 2 ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) = (𝐴 ++ 𝐵))
319, 10ercl2 7642 . . . 4 ((𝐴 𝐶𝐵 𝐷) → 𝐶 ∈ ( I ‘Word (𝐼 × 2𝑜)))
3231, 23eleqtrd 2690 . . 3 ((𝐴 𝐶𝐵 𝐷) → 𝐶 ∈ (Base‘𝑀))
339, 25ercl2 7642 . . . 4 ((𝐴 𝐶𝐵 𝐷) → 𝐷 ∈ ( I ‘Word (𝐼 × 2𝑜)))
3433, 23eleqtrd 2690 . . 3 ((𝐴 𝐶𝐵 𝐷) → 𝐷 ∈ (Base‘𝑀))
3519, 20, 28frmdadd 17215 . . 3 ((𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 + 𝐷) = (𝐶 ++ 𝐷))
3632, 34, 35syl2anc 691 . 2 ((𝐴 𝐶𝐵 𝐷) → (𝐶 + 𝐷) = (𝐶 ++ 𝐷))
377, 30, 363brtr4d 4615 1 ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {csn 4125  ⟨cop 4131  ⟨cotp 4133  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   × cxp 5036  ran crn 5039  Oncon0 5640  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1𝑜c1o 7440  2𝑜c2o 7441   Er wer 7626  0cc0 9815  1c1 9816   − cmin 10145  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148   splice csplice 13151  ⟨“cs2 13437  Basecbs 15695  +gcplusg 15768  freeMndcfrmd 17207   ~FG cefg 17942  freeGrpcfrgp 17943 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-rep 4699  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-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  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-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-frmd 17209  df-efg 17945 This theorem is referenced by:  frgp0  17996  frgpadd  17999
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