Theorem frgpuptinv 20036

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 = (invg‘𝐻)
frgpup.t	⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
frgpup.h	⊢ (𝜑 → 𝐻 ∈ Grp)
frgpup.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
frgpup.a	⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
frgpuptinv.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)

Assertion

Ref	Expression
frgpuptinv	⊢ ((𝜑 ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴)))

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.

Step	Hyp	Ref	Expression
1		elxp2 5682	. . 3 ⊢ (𝐴 ∈ (𝐼 × 2o) ↔ ∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩)
2		frgpuptinv.m	. . . . . . . . . 10 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
3	2	efgmval 19977	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
4	3	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
5	4	fveq2d 6920	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩))
6		df-ov 7454	. . . . . . 7 ⊢ (𝑎𝑇(1o ∖ 𝑏)) = (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩)
7	5, 6	eqtr4di 2816	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑎𝑇(1o ∖ 𝑏)))
8		elpri 4608	. . . . . . . . 9 ⊢ (𝑏 ∈ {∅, 1o} → (𝑏 = ∅ ∨ 𝑏 = 1o))
9		df2o3 8522	. . . . . . . . 9 ⊢ 2o = {∅, 1o}
10	8, 9	eleq2s 2882	. . . . . . . 8 ⊢ (𝑏 ∈ 2o → (𝑏 = ∅ ∨ 𝑏 = 1o))
11		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼)
12		1oelpr 8525	. . . . . . . . . . . . 13 ⊢ 1o ∈ {∅, 1o}
13	12, 9	eleqtrri 2863	. . . . . . . . . . . 12 ⊢ 1o ∈ 2o
14		1n0 8533	. . . . . . . . . . . . . . . 16 ⊢ 1o ≠ ∅
15		neeq1 3021	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1o → (𝑧 ≠ ∅ ↔ 1o ≠ ∅))
16	14, 15	mpbiri 261	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 1o → 𝑧 ≠ ∅)
17		ifnefalse 4497	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ≠ ∅ → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝑁‘(𝐹‘𝑦)))
18	16, 17	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝑁‘(𝐹‘𝑦)))
19		fveq2 6916	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑎 → (𝐹‘𝑦) = (𝐹‘𝑎))
20	19	fveq2d 6920	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → (𝑁‘(𝐹‘𝑦)) = (𝑁‘(𝐹‘𝑎)))
21	18, 20	sylan9eqr 2820	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑎 ∧ 𝑧 = 1o) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝑁‘(𝐹‘𝑎)))
22		frgpup.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
23		fvex 6929	. . . . . . . . . . . . 13 ⊢ (𝑁‘(𝐹‘𝑎)) ∈ V
24	21, 22, 23	ovmpoa 7607	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐼 ∧ 1o ∈ 2o) → (𝑎𝑇1o) = (𝑁‘(𝐹‘𝑎)))
25	11, 13, 24	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇1o) = (𝑁‘(𝐹‘𝑎)))
26		0ex 5267	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
27	26	prid1 4725	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ {∅, 1o}
28	27, 9	eleqtrri 2863	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 2o
29		iftrue 4491	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ∅ → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝑦))
30	29, 19	sylan9eqr 2820	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑎 ∧ 𝑧 = ∅) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝑎))
31		fvex 6929	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑎) ∈ V
32	30, 22, 31	ovmpoa 7607	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝑎𝑇∅) = (𝐹‘𝑎))
33	11, 28, 32	sylancl 598	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇∅) = (𝐹‘𝑎))
34	33	fveq2d 6920	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑁‘(𝑎𝑇∅)) = (𝑁‘(𝐹‘𝑎)))
35	25, 34	eqtr4d 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇1o) = (𝑁‘(𝑎𝑇∅)))
36		difeq2 4060	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → (1o ∖ 𝑏) = (1o ∖ ∅))
37		dif0 4330	. . . . . . . . . . . . 13 ⊢ (1o ∖ ∅) = 1o
38	36, 37	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (1o ∖ 𝑏) = 1o)
39	38	oveq2d 7467	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → (𝑎𝑇(1o ∖ 𝑏)) = (𝑎𝑇1o))
40		oveq2 7459	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (𝑎𝑇𝑏) = (𝑎𝑇∅))
41	40	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑎𝑇∅)))
42	39, 41	eqeq12d 2779	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → ((𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏)) ↔ (𝑎𝑇1o) = (𝑁‘(𝑎𝑇∅))))
43	35, 42	syl5ibrcom 250	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑏 = ∅ → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
44	35	fveq2d 6920	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑁‘(𝑎𝑇1o)) = (𝑁‘(𝑁‘(𝑎𝑇∅))))
45		frgpup.h	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ Grp)
46		frgpup.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
47	46	ffvelcdmda 7116	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝐹‘𝑎) ∈ 𝐵)
48	33, 47	eqeltrd 2864	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇∅) ∈ 𝐵)
49		frgpup.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐻)
50		frgpup.n	. . . . . . . . . . . . 13 ⊢ 𝑁 = (invg‘𝐻)
51	49, 50	grpinvinv 19266	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ Grp ∧ (𝑎𝑇∅) ∈ 𝐵) → (𝑁‘(𝑁‘(𝑎𝑇∅))) = (𝑎𝑇∅))
52	45, 48, 51	syl2an2r 698	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑁‘(𝑁‘(𝑎𝑇∅))) = (𝑎𝑇∅))
53	44, 52	eqtr2d 2799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇∅) = (𝑁‘(𝑎𝑇1o)))
54		difeq2 4060	. . . . . . . . . . . . 13 ⊢ (𝑏 = 1o → (1o ∖ 𝑏) = (1o ∖ 1o))
55		difid 4328	. . . . . . . . . . . . 13 ⊢ (1o ∖ 1o) = ∅
56	54, 55	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑏 = 1o → (1o ∖ 𝑏) = ∅)
57	56	oveq2d 7467	. . . . . . . . . . 11 ⊢ (𝑏 = 1o → (𝑎𝑇(1o ∖ 𝑏)) = (𝑎𝑇∅))
58		oveq2 7459	. . . . . . . . . . . 12 ⊢ (𝑏 = 1o → (𝑎𝑇𝑏) = (𝑎𝑇1o))
59	58	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝑏 = 1o → (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑎𝑇1o)))
60	57, 59	eqeq12d 2779	. . . . . . . . . 10 ⊢ (𝑏 = 1o → ((𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏)) ↔ (𝑎𝑇∅) = (𝑁‘(𝑎𝑇1o))))
61	53, 60	syl5ibrcom 250	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑏 = 1o → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
62	43, 61	jaod 873	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((𝑏 = ∅ ∨ 𝑏 = 1o) → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
63	10, 62	syl5 35	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑏 ∈ 2o → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
64	63	impr 460	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏)))
65	7, 64	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑁‘(𝑎𝑇𝑏)))
66		fveq2 6916	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
67		df-ov 7454	. . . . . . . 8 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
68	66, 67	eqtr4di 2816	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑎𝑀𝑏))
69	68	fveq2d 6920	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀‘𝐴)) = (𝑇‘(𝑎𝑀𝑏)))
70		fveq2 6916	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘𝐴) = (𝑇‘⟨𝑎, 𝑏⟩))
71		df-ov 7454	. . . . . . . 8 ⊢ (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩)
72	70, 71	eqtr4di 2816	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘𝐴) = (𝑎𝑇𝑏))
73	72	fveq2d 6920	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑁‘(𝑇‘𝐴)) = (𝑁‘(𝑎𝑇𝑏)))
74	69, 73	eqeq12d 2779	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴)) ↔ (𝑇‘(𝑎𝑀𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
75	65, 74	syl5ibrcom 250	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴))))
76	75	rexlimdvva 3223	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴))))
77	1, 76	biimtrid 245	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝐼 × 2o) → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴))))
78	77	imp 412	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   ∖ cdif 4048  ∅c0 4281  ifcif 4485  {cpr 4590  ⟨cop 4840   × cxp 5656  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1_oc1o 8507  2_oc2o 8508  Basecbs 17431  Grpcgrp 19194  inv_gcminusg 19195

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1o 8514  df-2o 8515  df-0g 17662  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-grp 19197  df-minusg 19198

This theorem is used by:  frgpuplem  20037