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Theorem frgpuptinv 18230
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 5166 . . 3 (𝐴 ∈ (𝐼 × 2𝑜) ↔ ∃𝑎𝐼𝑏 ∈ 2𝑜 𝐴 = ⟨𝑎, 𝑏⟩)
2 frgpuptinv.m . . . . . . . . . 10 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
32efgmval 18171 . . . . . . . . 9 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜𝑏)⟩)
43adantl 481 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜𝑏)⟩)
54fveq2d 6233 . . . . . . 7 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑇‘⟨𝑎, (1𝑜𝑏)⟩))
6 df-ov 6693 . . . . . . 7 (𝑎𝑇(1𝑜𝑏)) = (𝑇‘⟨𝑎, (1𝑜𝑏)⟩)
75, 6syl6eqr 2703 . . . . . 6 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑎𝑇(1𝑜𝑏)))
8 elpri 4230 . . . . . . . . 9 (𝑏 ∈ {∅, 1𝑜} → (𝑏 = ∅ ∨ 𝑏 = 1𝑜))
9 df2o3 7618 . . . . . . . . 9 2𝑜 = {∅, 1𝑜}
108, 9eleq2s 2748 . . . . . . . 8 (𝑏 ∈ 2𝑜 → (𝑏 = ∅ ∨ 𝑏 = 1𝑜))
11 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑎𝐼) → 𝑎𝐼)
12 1on 7612 . . . . . . . . . . . . . . 15 1𝑜 ∈ On
1312elexi 3244 . . . . . . . . . . . . . 14 1𝑜 ∈ V
1413prid2 4330 . . . . . . . . . . . . 13 1𝑜 ∈ {∅, 1𝑜}
1514, 9eleqtrri 2729 . . . . . . . . . . . 12 1𝑜 ∈ 2𝑜
16 1n0 7620 . . . . . . . . . . . . . . . 16 1𝑜 ≠ ∅
17 neeq1 2885 . . . . . . . . . . . . . . . 16 (𝑧 = 1𝑜 → (𝑧 ≠ ∅ ↔ 1𝑜 ≠ ∅))
1816, 17mpbiri 248 . . . . . . . . . . . . . . 15 (𝑧 = 1𝑜𝑧 ≠ ∅)
19 ifnefalse 4131 . . . . . . . . . . . . . . 15 (𝑧 ≠ ∅ → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝑁‘(𝐹𝑦)))
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝑧 = 1𝑜 → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝑁‘(𝐹𝑦)))
21 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (𝐹𝑦) = (𝐹𝑎))
2221fveq2d 6233 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → (𝑁‘(𝐹𝑦)) = (𝑁‘(𝐹𝑎)))
2320, 22sylan9eqr 2707 . . . . . . . . . . . . 13 ((𝑦 = 𝑎𝑧 = 1𝑜) → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝑁‘(𝐹𝑎)))
24 frgpup.t . . . . . . . . . . . . 13 𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
25 fvex 6239 . . . . . . . . . . . . 13 (𝑁‘(𝐹𝑎)) ∈ V
2623, 24, 25ovmpt2a 6833 . . . . . . . . . . . 12 ((𝑎𝐼 ∧ 1𝑜 ∈ 2𝑜) → (𝑎𝑇1𝑜) = (𝑁‘(𝐹𝑎)))
2711, 15, 26sylancl 695 . . . . . . . . . . 11 ((𝜑𝑎𝐼) → (𝑎𝑇1𝑜) = (𝑁‘(𝐹𝑎)))
28 0ex 4823 . . . . . . . . . . . . . . 15 ∅ ∈ V
2928prid1 4329 . . . . . . . . . . . . . 14 ∅ ∈ {∅, 1𝑜}
3029, 9eleqtrri 2729 . . . . . . . . . . . . 13 ∅ ∈ 2𝑜
31 iftrue 4125 . . . . . . . . . . . . . . 15 (𝑧 = ∅ → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝐹𝑦))
3231, 21sylan9eqr 2707 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑧 = ∅) → if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))) = (𝐹𝑎))
33 fvex 6239 . . . . . . . . . . . . . 14 (𝐹𝑎) ∈ V
3432, 24, 33ovmpt2a 6833 . . . . . . . . . . . . 13 ((𝑎𝐼 ∧ ∅ ∈ 2𝑜) → (𝑎𝑇∅) = (𝐹𝑎))
3511, 30, 34sylancl 695 . . . . . . . . . . . 12 ((𝜑𝑎𝐼) → (𝑎𝑇∅) = (𝐹𝑎))
3635fveq2d 6233 . . . . . . . . . . 11 ((𝜑𝑎𝐼) → (𝑁‘(𝑎𝑇∅)) = (𝑁‘(𝐹𝑎)))
3727, 36eqtr4d 2688 . . . . . . . . . 10 ((𝜑𝑎𝐼) → (𝑎𝑇1𝑜) = (𝑁‘(𝑎𝑇∅)))
38 difeq2 3755 . . . . . . . . . . . . 13 (𝑏 = ∅ → (1𝑜𝑏) = (1𝑜 ∖ ∅))
39 dif0 3983 . . . . . . . . . . . . 13 (1𝑜 ∖ ∅) = 1𝑜
4038, 39syl6eq 2701 . . . . . . . . . . . 12 (𝑏 = ∅ → (1𝑜𝑏) = 1𝑜)
4140oveq2d 6706 . . . . . . . . . . 11 (𝑏 = ∅ → (𝑎𝑇(1𝑜𝑏)) = (𝑎𝑇1𝑜))
42 oveq2 6698 . . . . . . . . . . . 12 (𝑏 = ∅ → (𝑎𝑇𝑏) = (𝑎𝑇∅))
4342fveq2d 6233 . . . . . . . . . . 11 (𝑏 = ∅ → (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑎𝑇∅)))
4441, 43eqeq12d 2666 . . . . . . . . . 10 (𝑏 = ∅ → ((𝑎𝑇(1𝑜𝑏)) = (𝑁‘(𝑎𝑇𝑏)) ↔ (𝑎𝑇1𝑜) = (𝑁‘(𝑎𝑇∅))))
4537, 44syl5ibrcom 237 . . . . . . . . 9 ((𝜑𝑎𝐼) → (𝑏 = ∅ → (𝑎𝑇(1𝑜𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
4637fveq2d 6233 . . . . . . . . . . 11 ((𝜑𝑎𝐼) → (𝑁‘(𝑎𝑇1𝑜)) = (𝑁‘(𝑁‘(𝑎𝑇∅))))
47 frgpup.h . . . . . . . . . . . . 13 (𝜑𝐻 ∈ Grp)
4847adantr 480 . . . . . . . . . . . 12 ((𝜑𝑎𝐼) → 𝐻 ∈ Grp)
49 frgpup.a . . . . . . . . . . . . . 14 (𝜑𝐹:𝐼𝐵)
5049ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑𝑎𝐼) → (𝐹𝑎) ∈ 𝐵)
5135, 50eqeltrd 2730 . . . . . . . . . . . 12 ((𝜑𝑎𝐼) → (𝑎𝑇∅) ∈ 𝐵)
52 frgpup.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐻)
53 frgpup.n . . . . . . . . . . . . 13 𝑁 = (invg𝐻)
5452, 53grpinvinv 17529 . . . . . . . . . . . 12 ((𝐻 ∈ Grp ∧ (𝑎𝑇∅) ∈ 𝐵) → (𝑁‘(𝑁‘(𝑎𝑇∅))) = (𝑎𝑇∅))
5548, 51, 54syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑎𝐼) → (𝑁‘(𝑁‘(𝑎𝑇∅))) = (𝑎𝑇∅))
5646, 55eqtr2d 2686 . . . . . . . . . 10 ((𝜑𝑎𝐼) → (𝑎𝑇∅) = (𝑁‘(𝑎𝑇1𝑜)))
57 difeq2 3755 . . . . . . . . . . . . 13 (𝑏 = 1𝑜 → (1𝑜𝑏) = (1𝑜 ∖ 1𝑜))
58 difid 3981 . . . . . . . . . . . . 13 (1𝑜 ∖ 1𝑜) = ∅
5957, 58syl6eq 2701 . . . . . . . . . . . 12 (𝑏 = 1𝑜 → (1𝑜𝑏) = ∅)
6059oveq2d 6706 . . . . . . . . . . 11 (𝑏 = 1𝑜 → (𝑎𝑇(1𝑜𝑏)) = (𝑎𝑇∅))
61 oveq2 6698 . . . . . . . . . . . 12 (𝑏 = 1𝑜 → (𝑎𝑇𝑏) = (𝑎𝑇1𝑜))
6261fveq2d 6233 . . . . . . . . . . 11 (𝑏 = 1𝑜 → (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑎𝑇1𝑜)))
6360, 62eqeq12d 2666 . . . . . . . . . 10 (𝑏 = 1𝑜 → ((𝑎𝑇(1𝑜𝑏)) = (𝑁‘(𝑎𝑇𝑏)) ↔ (𝑎𝑇∅) = (𝑁‘(𝑎𝑇1𝑜))))
6456, 63syl5ibrcom 237 . . . . . . . . 9 ((𝜑𝑎𝐼) → (𝑏 = 1𝑜 → (𝑎𝑇(1𝑜𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
6545, 64jaod 394 . . . . . . . 8 ((𝜑𝑎𝐼) → ((𝑏 = ∅ ∨ 𝑏 = 1𝑜) → (𝑎𝑇(1𝑜𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
6610, 65syl5 34 . . . . . . 7 ((𝜑𝑎𝐼) → (𝑏 ∈ 2𝑜 → (𝑎𝑇(1𝑜𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
6766impr 648 . . . . . 6 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑎𝑇(1𝑜𝑏)) = (𝑁‘(𝑎𝑇𝑏)))
687, 67eqtrd 2685 . . . . 5 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑁‘(𝑎𝑇𝑏)))
69 fveq2 6229 . . . . . . . 8 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
70 df-ov 6693 . . . . . . . 8 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
7169, 70syl6eqr 2703 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀𝐴) = (𝑎𝑀𝑏))
7271fveq2d 6233 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀𝐴)) = (𝑇‘(𝑎𝑀𝑏)))
73 fveq2 6229 . . . . . . . 8 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇𝐴) = (𝑇‘⟨𝑎, 𝑏⟩))
74 df-ov 6693 . . . . . . . 8 (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩)
7573, 74syl6eqr 2703 . . . . . . 7 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇𝐴) = (𝑎𝑇𝑏))
7675fveq2d 6233 . . . . . 6 (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑁‘(𝑇𝐴)) = (𝑁‘(𝑎𝑇𝑏)))
7772, 76eqeq12d 2666 . . . . 5 (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴)) ↔ (𝑇‘(𝑎𝑀𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
7868, 77syl5ibrcom 237 . . . 4 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴))))
7978rexlimdvva 3067 . . 3 (𝜑 → (∃𝑎𝐼𝑏 ∈ 2𝑜 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴))))
801, 79syl5bi 232 . 2 (𝜑 → (𝐴 ∈ (𝐼 × 2𝑜) → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴))))
8180imp 444 1 ((𝜑𝐴 ∈ (𝐼 × 2𝑜)) → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wrex 2942  cdif 3604  c0 3948  ifcif 4119  {cpr 4212  cop 4216   × cxp 5141  Oncon0 5761  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1𝑜c1o 7598  2𝑜c2o 7599  Basecbs 15904  Grpcgrp 17469  invgcminusg 17470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1o 7605  df-2o 7606  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473
This theorem is referenced by:  frgpuplem  18231
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