Theorem frgpuptinv 19813

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 5724	. . 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 19754	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
4	3	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
5	4	fveq2d 6924	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩))
6		df-ov 7451	. . . . . . 7 ⊢ (𝑎𝑇(1o ∖ 𝑏)) = (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩)
7	5, 6	eqtr4di 2798	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑎𝑇(1o ∖ 𝑏)))
8		elpri 4671	. . . . . . . . 9 ⊢ (𝑏 ∈ {∅, 1o} → (𝑏 = ∅ ∨ 𝑏 = 1o))
9		df2o3 8530	. . . . . . . . 9 ⊢ 2o = {∅, 1o}
10	8, 9	eleq2s 2862	. . . . . . . 8 ⊢ (𝑏 ∈ 2o → (𝑏 = ∅ ∨ 𝑏 = 1o))
11		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → 𝑎 ∈ 𝐼)
12		1oex 8532	. . . . . . . . . . . . . 14 ⊢ 1o ∈ V
13	12	prid2 4788	. . . . . . . . . . . . 13 ⊢ 1o ∈ {∅, 1o}
14	13, 9	eleqtrri 2843	. . . . . . . . . . . 12 ⊢ 1o ∈ 2o
15		1n0 8544	. . . . . . . . . . . . . . . 16 ⊢ 1o ≠ ∅
16		neeq1 3009	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1o → (𝑧 ≠ ∅ ↔ 1o ≠ ∅))
17	15, 16	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 1o → 𝑧 ≠ ∅)
18		ifnefalse 4560	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ≠ ∅ → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝑁‘(𝐹‘𝑦)))
19	17, 18	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝑁‘(𝐹‘𝑦)))
20		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑎 → (𝐹‘𝑦) = (𝐹‘𝑎))
21	20	fveq2d 6924	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → (𝑁‘(𝐹‘𝑦)) = (𝑁‘(𝐹‘𝑎)))
22	19, 21	sylan9eqr 2802	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑎 ∧ 𝑧 = 1o) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝑁‘(𝐹‘𝑎)))
23		frgpup.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
24		fvex 6933	. . . . . . . . . . . . 13 ⊢ (𝑁‘(𝐹‘𝑎)) ∈ V
25	22, 23, 24	ovmpoa 7605	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐼 ∧ 1o ∈ 2o) → (𝑎𝑇1o) = (𝑁‘(𝐹‘𝑎)))
26	11, 14, 25	sylancl 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇1o) = (𝑁‘(𝐹‘𝑎)))
27		0ex 5325	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
28	27	prid1 4787	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ {∅, 1o}
29	28, 9	eleqtrri 2843	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 2o
30		iftrue 4554	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ∅ → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝑦))
31	30, 20	sylan9eqr 2802	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑎 ∧ 𝑧 = ∅) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝑎))
32		fvex 6933	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑎) ∈ V
33	31, 23, 32	ovmpoa 7605	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝑎𝑇∅) = (𝐹‘𝑎))
34	11, 29, 33	sylancl 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇∅) = (𝐹‘𝑎))
35	34	fveq2d 6924	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑁‘(𝑎𝑇∅)) = (𝑁‘(𝐹‘𝑎)))
36	26, 35	eqtr4d 2783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇1o) = (𝑁‘(𝑎𝑇∅)))
37		difeq2 4143	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → (1o ∖ 𝑏) = (1o ∖ ∅))
38		dif0 4400	. . . . . . . . . . . . 13 ⊢ (1o ∖ ∅) = 1o
39	37, 38	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (1o ∖ 𝑏) = 1o)
40	39	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → (𝑎𝑇(1o ∖ 𝑏)) = (𝑎𝑇1o))
41		oveq2 7456	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (𝑎𝑇𝑏) = (𝑎𝑇∅))
42	41	fveq2d 6924	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑎𝑇∅)))
43	40, 42	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → ((𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏)) ↔ (𝑎𝑇1o) = (𝑁‘(𝑎𝑇∅))))
44	36, 43	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑏 = ∅ → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
45	36	fveq2d 6924	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑁‘(𝑎𝑇1o)) = (𝑁‘(𝑁‘(𝑎𝑇∅))))
46		frgpup.h	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ Grp)
47		frgpup.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
48	47	ffvelcdmda 7118	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝐹‘𝑎) ∈ 𝐵)
49	34, 48	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇∅) ∈ 𝐵)
50		frgpup.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐻)
51		frgpup.n	. . . . . . . . . . . . 13 ⊢ 𝑁 = (invg‘𝐻)
52	50, 51	grpinvinv 19045	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ Grp ∧ (𝑎𝑇∅) ∈ 𝐵) → (𝑁‘(𝑁‘(𝑎𝑇∅))) = (𝑎𝑇∅))
53	46, 49, 52	syl2an2r 684	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑁‘(𝑁‘(𝑎𝑇∅))) = (𝑎𝑇∅))
54	45, 53	eqtr2d 2781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑎𝑇∅) = (𝑁‘(𝑎𝑇1o)))
55		difeq2 4143	. . . . . . . . . . . . 13 ⊢ (𝑏 = 1o → (1o ∖ 𝑏) = (1o ∖ 1o))
56		difid 4398	. . . . . . . . . . . . 13 ⊢ (1o ∖ 1o) = ∅
57	55, 56	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ (𝑏 = 1o → (1o ∖ 𝑏) = ∅)
58	57	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑏 = 1o → (𝑎𝑇(1o ∖ 𝑏)) = (𝑎𝑇∅))
59		oveq2 7456	. . . . . . . . . . . 12 ⊢ (𝑏 = 1o → (𝑎𝑇𝑏) = (𝑎𝑇1o))
60	59	fveq2d 6924	. . . . . . . . . . 11 ⊢ (𝑏 = 1o → (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑎𝑇1o)))
61	58, 60	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑏 = 1o → ((𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏)) ↔ (𝑎𝑇∅) = (𝑁‘(𝑎𝑇1o))))
62	54, 61	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑏 = 1o → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
63	44, 62	jaod 858	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((𝑏 = ∅ ∨ 𝑏 = 1o) → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
64	10, 63	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (𝑏 ∈ 2o → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
65	64	impr 454	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑎𝑇(1o ∖ 𝑏)) = (𝑁‘(𝑎𝑇𝑏)))
66	7, 65	eqtrd 2780	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘(𝑎𝑀𝑏)) = (𝑁‘(𝑎𝑇𝑏)))
67		fveq2 6920	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑀‘⟨𝑎, 𝑏⟩))
68		df-ov 7451	. . . . . . . 8 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
69	67, 68	eqtr4di 2798	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝐴) = (𝑎𝑀𝑏))
70	69	fveq2d 6924	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀‘𝐴)) = (𝑇‘(𝑎𝑀𝑏)))
71		fveq2 6920	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘𝐴) = (𝑇‘⟨𝑎, 𝑏⟩))
72		df-ov 7451	. . . . . . . 8 ⊢ (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩)
73	71, 72	eqtr4di 2798	. . . . . . 7 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘𝐴) = (𝑎𝑇𝑏))
74	73	fveq2d 6924	. . . . . 6 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑁‘(𝑇‘𝐴)) = (𝑁‘(𝑎𝑇𝑏)))
75	70, 74	eqeq12d 2756	. . . . 5 ⊢ (𝐴 = ⟨𝑎, 𝑏⟩ → ((𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴)) ↔ (𝑇‘(𝑎𝑀𝑏)) = (𝑁‘(𝑎𝑇𝑏))))
76	66, 75	syl5ibrcom 247	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴))))
77	76	rexlimdvva 3219	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 2o 𝐴 = ⟨𝑎, 𝑏⟩ → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴))))
78	1, 77	biimtrid 242	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝐼 × 2o) → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴))))
79	78	imp 406	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076   ∖ cdif 3973  ∅c0 4352  ifcif 4548  {cpr 4650  ⟨cop 4654   × cxp 5698  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1_oc1o 8515  2_oc2o 8516  Basecbs 17258  Grpcgrp 18973  inv_gcminusg 18974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1o 8522  df-2o 8523  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977

This theorem is referenced by:  frgpuplem  19814