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Theorem frgpuptinv 18893
 Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
frgpup.b 𝐵 = (Base‘𝐻)
frgpup.n 𝑁 = (invg𝐻)
frgpup.t 𝑇 = (𝑦𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
frgpup.h (𝜑𝐻 ∈ Grp)
frgpup.i (𝜑𝐼𝑉)
frgpup.a (𝜑𝐹:𝐼𝐵)
frgpuptinv.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
Assertion
Ref Expression
frgpuptinv ((𝜑𝐴 ∈ (𝐼 × 2o)) → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴)))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐹,𝑧   𝑦,𝑁,𝑧   𝑦,𝐵,𝑧   𝜑,𝑦,𝑧   𝑦,𝐼,𝑧
Allowed substitution hints:   𝑇(𝑦,𝑧)   𝐻(𝑦,𝑧)   𝑀(𝑦,𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem frgpuptinv
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 5566 . . 3 (𝐴 ∈ (𝐼 × 2o) ↔ ∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩)
2 frgpuptinv.m . . . . . . . . . 10 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
32efgmval 18834 . . . . . . . . 9 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o𝑏)⟩)
43adantl 485 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑎𝑀𝑏) = ⟨𝑎, (1o𝑏)⟩)
54fveq2d 6662 . . . . . . 7 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑇‘⟨𝑎, (1o𝑏)⟩))
6 df-ov 7148 . . . . . . 7 (𝑎𝑇(1o𝑏)) = (𝑇‘⟨𝑎, (1o𝑏)⟩)
75, 6syl6eqr 2877 . . . . . 6 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑎𝑇(1o𝑏)))
8 elpri 4571 . . . . . . . . 9 (𝑏 ∈ {∅, 1o} → (𝑏 = ∅ ∨ 𝑏 = 1o))
9 df2o3 8107 . . . . . . . . 9 2o = {∅, 1o}
108, 9eleq2s 2934 . . . . . . . 8 (𝑏 ∈ 2o → (𝑏 = ∅ ∨ 𝑏 = 1o))
11 simpr 488 . . . . . . . . . . . 12 ((𝜑𝑎𝐼) → 𝑎𝐼)
12 1oex 8100 . . . . . . . . . . . . . 14 1o ∈ V
1312prid2 4683 . . . . . . . . . . . . 13 1o ∈ {∅, 1o}
1413, 9eleqtrri 2915 . . . . . . . . . . . 12 1o ∈ 2o
15 1n0 8109 . . . . . . . . . . . . . . . 16 1o ≠ ∅
16 neeq1 3076 . . . . . . . . . . . . . . . 16 (𝑧 = 1o → (𝑧 ≠ ∅ ↔ 1o ≠ ∅))
1715, 16mpbiri 261 . . . . . . . . . . . . . . 15 (𝑧 = 1o𝑧 ≠ ∅)
18 ifnefalse 4461 . . . . . . . . . . . . . . 15 (𝑧 ≠ ∅ → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝑁‘(𝐹𝑦)))
1917, 18syl 17 . . . . . . . . . . . . . 14 (𝑧 = 1o → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝑁‘(𝐹𝑦)))
20 fveq2 6658 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (𝐹𝑦) = (𝐹𝑎))
2120fveq2d 6662 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → (𝑁‘(𝐹𝑦)) = (𝑁‘(𝐹𝑎)))
2219, 21sylan9eqr 2881 . . . . . . . . . . . . 13 ((𝑦 = 𝑎𝑧 = 1o) → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝑁‘(𝐹𝑎)))
23 frgpup.t . . . . . . . . . . . . 13 𝑇 = (𝑦𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
24 fvex 6671 . . . . . . . . . . . . 13 (𝑁‘(𝐹𝑎)) ∈ V
2522, 23, 24ovmpoa 7294 . . . . . . . . . . . 12 ((𝑎𝐼 ∧ 1o ∈ 2o) → (𝑎𝑇1o) = (𝑁‘(𝐹𝑎)))
2611, 14, 25sylancl 589 . . . . . . . . . . 11 ((𝜑𝑎𝐼) → (𝑎𝑇1o) = (𝑁‘(𝐹𝑎)))
27 0ex 5197 . . . . . . . . . . . . . . 15 ∅ ∈ V
2827prid1 4682 . . . . . . . . . . . . . 14 ∅ ∈ {∅, 1o}
2928, 9eleqtrri 2915 . . . . . . . . . . . . 13 ∅ ∈ 2o
30 iftrue 4455 . . . . . . . . . . . . . . 15 (𝑧 = ∅ → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝐹𝑦))
3130, 20sylan9eqr 2881 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑧 = ∅) → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝐹𝑎))
32 fvex 6671 . . . . . . . . . . . . . 14 (𝐹𝑎) ∈ V
3331, 23, 32ovmpoa 7294 . . . . . . . . . . . . 13 ((𝑎𝐼 ∧ ∅ ∈ 2o) → (𝑎𝑇∅) = (𝐹𝑎))
3411, 29, 33sylancl 589 . . . . . . . . . . . 12 ((𝜑𝑎𝐼) → (𝑎𝑇∅) = (𝐹𝑎))
3534fveq2d 6662 . . . . . . . . . . 11 ((𝜑𝑎𝐼) → (𝑁‘(𝑎𝑇∅)) = (𝑁‘(𝐹𝑎)))
3626, 35eqtr4d 2862 . . . . . . . . . 10 ((𝜑𝑎𝐼) → (𝑎𝑇1o) = (𝑁‘(𝑎𝑇∅)))
37 difeq2 4078 . . . . . . . . . . . . 13 (𝑏 = ∅ → (1o𝑏) = (1o ∖ ∅))
38 dif0 4314 . . . . . . . . . . . . 13 (1o ∖ ∅) = 1o
3937, 38syl6eq 2875 . . . . . . . . . . . 12 (𝑏 = ∅ → (1o𝑏) = 1o)
4039oveq2d 7161 . . . . . . . . . . 11 (𝑏 = ∅ → (𝑎𝑇(1o𝑏)) = (𝑎𝑇1o))
41 oveq2 7153 . . . . . . . . . . . 12 (𝑏 = ∅ → (𝑎𝑇𝑏) = (𝑎𝑇∅))
4241fveq2d 6662 . . . . . . . . . . 11 (𝑏 = ∅ → (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑎𝑇∅)))
4340, 42eqeq12d 2840 . . . . . . . . . 10 (𝑏 = ∅ → ((𝑎𝑇(1o𝑏)) = (𝑁‘(𝑎𝑇𝑏)) ↔ (𝑎𝑇1o) = (𝑁‘(𝑎𝑇∅))))
4436, 43syl5ibrcom 250 . . . . . . . . 9 ((𝜑𝑎𝐼) → (𝑏 = ∅ → (𝑎𝑇(1o𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
4536fveq2d 6662 . . . . . . . . . . 11 ((𝜑𝑎𝐼) → (𝑁‘(𝑎𝑇1o)) = (𝑁‘(𝑁‘(𝑎𝑇∅))))
46 frgpup.h . . . . . . . . . . . 12 (𝜑𝐻 ∈ Grp)
47 frgpup.a . . . . . . . . . . . . . 14 (𝜑𝐹:𝐼𝐵)
4847ffvelrnda 6839 . . . . . . . . . . . . 13 ((𝜑𝑎𝐼) → (𝐹𝑎) ∈ 𝐵)
4934, 48eqeltrd 2916 . . . . . . . . . . . 12 ((𝜑𝑎𝐼) → (𝑎𝑇∅) ∈ 𝐵)
50 frgpup.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐻)
51 frgpup.n . . . . . . . . . . . . 13 𝑁 = (invg𝐻)
5250, 51grpinvinv 18162 . . . . . . . . . . . 12 ((𝐻 ∈ Grp ∧ (𝑎𝑇∅) ∈ 𝐵) → (𝑁‘(𝑁‘(𝑎𝑇∅))) = (𝑎𝑇∅))
5346, 49, 52syl2an2r 684 . . . . . . . . . . 11 ((𝜑𝑎𝐼) → (𝑁‘(𝑁‘(𝑎𝑇∅))) = (𝑎𝑇∅))
5445, 53eqtr2d 2860 . . . . . . . . . 10 ((𝜑𝑎𝐼) → (𝑎𝑇∅) = (𝑁‘(𝑎𝑇1o)))
55 difeq2 4078 . . . . . . . . . . . . 13 (𝑏 = 1o → (1o𝑏) = (1o ∖ 1o))
56 difid 4312 . . . . . . . . . . . . 13 (1o ∖ 1o) = ∅
5755, 56syl6eq 2875 . . . . . . . . . . . 12 (𝑏 = 1o → (1o𝑏) = ∅)
5857oveq2d 7161 . . . . . . . . . . 11 (𝑏 = 1o → (𝑎𝑇(1o𝑏)) = (𝑎𝑇∅))
59 oveq2 7153 . . . . . . . . . . . 12 (𝑏 = 1o → (𝑎𝑇𝑏) = (𝑎𝑇1o))
6059fveq2d 6662 . . . . . . . . . . 11 (𝑏 = 1o → (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑎𝑇1o)))
6158, 60eqeq12d 2840 . . . . . . . . . 10 (𝑏 = 1o → ((𝑎𝑇(1o𝑏)) = (𝑁‘(𝑎𝑇𝑏)) ↔ (𝑎𝑇∅) = (𝑁‘(𝑎𝑇1o))))
6254, 61syl5ibrcom 250 . . . . . . . . 9 ((𝜑𝑎𝐼) → (𝑏 = 1o → (𝑎𝑇(1o𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
6344, 62jaod 856 . . . . . . . 8 ((𝜑𝑎𝐼) → ((𝑏 = ∅ ∨ 𝑏 = 1o) → (𝑎𝑇(1o𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
6410, 63syl5 34 . . . . . . 7 ((𝜑𝑎𝐼) → (𝑏 ∈ 2o → (𝑎𝑇(1o𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
6564impr 458 . . . . . 6 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑎𝑇(1o𝑏)) = (𝑁‘(𝑎𝑇𝑏)))
667, 65eqtrd 2859 . . . . 5 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑁‘(𝑎𝑇𝑏)))
67 fveq2 6658 . . . . . . . 8 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
68 df-ov 7148 . . . . . . . 8 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
6967, 68syl6eqr 2877 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑎𝑀𝑏))
7069fveq2d 6662 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀𝐴)) = (𝑇‘(𝑎𝑀𝑏)))
71 fveq2 6658 . . . . . . . 8 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇𝐴) = (𝑇‘⟨𝑎, 𝑏⟩))
72 df-ov 7148 . . . . . . . 8 (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩)
7371, 72syl6eqr 2877 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇𝐴) = (𝑎𝑇𝑏))
7473fveq2d 6662 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑁‘(𝑇𝐴)) = (𝑁‘(𝑎𝑇𝑏)))
7570, 74eqeq12d 2840 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴)) ↔ (𝑇‘(𝑎𝑀𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
7666, 75syl5ibrcom 250 . . . 4 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴))))
7776rexlimdvva 3287 . . 3 (𝜑 → (∃𝑎𝐼𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴))))
781, 77syl5bi 245 . 2 (𝜑 → (𝐴 ∈ (𝐼 × 2o) → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴))))
7978imp 410 1 ((𝜑𝐴 ∈ (𝐼 × 2o)) → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∃wrex 3134   ∖ cdif 3916  ∅c0 4275  ifcif 4449  {cpr 4551  ⟨cop 4555   × cxp 5540  ⟶wf 6339  ‘cfv 6343  (class class class)co 7145   ∈ cmpo 7147  1oc1o 8085  2oc2o 8086  Basecbs 16479  Grpcgrp 18099  invgcminusg 18100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1o 8092  df-2o 8093  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-grp 18102  df-minusg 18103 This theorem is referenced by:  frgpuplem  18894
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