Theorem frmdup1 18924

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, 27-Sep-2015.)

Hypotheses

Ref	Expression
frmdup.m	⊢ 𝑀 = (freeMnd‘𝐼)
frmdup.b	⊢ 𝐵 = (Base‘𝐺)
frmdup.e	⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))
frmdup.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
frmdup.i	⊢ (𝜑 → 𝐼 ∈ 𝑋)
frmdup.a	⊢ (𝜑 → 𝐴:𝐼⟶𝐵)

Assertion

Ref	Expression
frmdup1	⊢ (𝜑 → 𝐸 ∈ (𝑀 MndHom 𝐺))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐺   𝜑,𝑥   𝑥,𝐼

Allowed substitution hints:   𝐸(𝑥)   𝑀(𝑥)   𝑋(𝑥)

Proof of Theorem frmdup1

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frmdup.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑋)
2		frmdup.m	. . . 4 ⊢ 𝑀 = (freeMnd‘𝐼)
3	2	frmdmnd 18919	. . 3 ⊢ (𝐼 ∈ 𝑋 → 𝑀 ∈ Mnd)
4	1, 3	syl 18	. 2 ⊢ (𝜑 → 𝑀 ∈ Mnd)
5		frmdup.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
6	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝐺 ∈ Mnd)
7		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝑥 ∈ Word 𝐼)
8		frmdup.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:𝐼⟶𝐵)
9	8	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝐴:𝐼⟶𝐵)
10		wrdco 14870	. . . . . . 7 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 𝑥) ∈ Word 𝐵)
11	7, 9, 10	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → (𝐴 ∘ 𝑥) ∈ Word 𝐵)
12		frmdup.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
13	12	gsumwcl 18899	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∘ 𝑥) ∈ Word 𝐵) → (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ 𝐵)
14	6, 11, 13	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ 𝐵)
15		frmdup.e	. . . . 5 ⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))
16	14, 15	fmptd 7111	. . . 4 ⊢ (𝜑 → 𝐸:Word 𝐼⟶𝐵)
17		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
18	2, 17	frmdbas 18912	. . . . . 6 ⊢ (𝐼 ∈ 𝑋 → (Base‘𝑀) = Word 𝐼)
19	1, 18	syl 18	. . . . 5 ⊢ (𝜑 → (Base‘𝑀) = Word 𝐼)
20	19	feq2d 6691	. . . 4 ⊢ (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ↔ 𝐸:Word 𝐼⟶𝐵))
21	16, 20	mpbird 260	. . 3 ⊢ (𝜑 → 𝐸:(Base‘𝑀)⟶𝐵)
22	2, 17	frmdelbas 18913	. . . . . . . . 9 ⊢ (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
23	22	ad2antrl 740	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
24	2, 17	frmdelbas 18913	. . . . . . . . 9 ⊢ (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
25	24	ad2antll 741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
26	8	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐴:𝐼⟶𝐵)
27		ccatco 14874	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧)))
28	23, 25, 26, 27	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧)))
29	28	oveq2d 7428	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = (𝐺 Σg ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))))
30	5	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐺 ∈ Mnd)
31		wrdco 14870	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 𝑦) ∈ Word 𝐵)
32	23, 26, 31	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ 𝑦) ∈ Word 𝐵)
33		wrdco 14870	. . . . . . . 8 ⊢ ((𝑧 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 𝑧) ∈ Word 𝐵)
34	25, 26, 33	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ 𝑧) ∈ Word 𝐵)
35		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
36	12, 35	gsumccat 18901	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∘ 𝑦) ∈ Word 𝐵 ∧ (𝐴 ∘ 𝑧) ∈ Word 𝐵) → (𝐺 Σg ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
37	30, 32, 34, 36	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
38	29, 37	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
39		eqid 2763	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
40	2, 17, 39	frmdadd 18915	. . . . . . . 8 ⊢ ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
41	40	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
42	41	fveq2d 6887	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g‘𝑀)𝑧)) = (𝐸‘(𝑦 ++ 𝑧)))
43		ccatcl 14613	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
44	23, 25, 43	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
45		coeq2 5846	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ++ 𝑧) → (𝐴 ∘ 𝑥) = (𝐴 ∘ (𝑦 ++ 𝑧)))
46	45	oveq2d 7428	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ++ 𝑧) → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
47		ovex 7445	. . . . . . . 8 ⊢ (𝐺 Σg (𝐴 ∘ 𝑥)) ∈ V
48	46, 15, 47	fvmpt3i 6997	. . . . . . 7 ⊢ ((𝑦 ++ 𝑧) ∈ Word 𝐼 → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
49	44, 48	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
50	42, 49	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g‘𝑀)𝑧)) = (𝐺 Σg (𝐴 ∘ (𝑦 ++ 𝑧))))
51		coeq2 5846	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 𝑦))
52	51	oveq2d 7428	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ 𝑦)))
53	52, 15, 47	fvmpt3i 6997	. . . . . . 7 ⊢ (𝑦 ∈ Word 𝐼 → (𝐸‘𝑦) = (𝐺 Σg (𝐴 ∘ 𝑦)))
54		coeq2 5846	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 𝑧))
55	54	oveq2d 7428	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg (𝐴 ∘ 𝑧)))
56	55, 15, 47	fvmpt3i 6997	. . . . . . 7 ⊢ (𝑧 ∈ Word 𝐼 → (𝐸‘𝑧) = (𝐺 Σg (𝐴 ∘ 𝑧)))
57	53, 56	oveqan12d 7431	. . . . . 6 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
58	23, 25, 57	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)) = ((𝐺 Σg (𝐴 ∘ 𝑦))(+g‘𝐺)(𝐺 Σg (𝐴 ∘ 𝑧))))
59	38, 50, 58	3eqtr4d 2808	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)))
60	59	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)))
61		wrd0 14578	. . . 4 ⊢ ∅ ∈ Word 𝐼
62		coeq2 5846	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝐴 ∘ 𝑥) = (𝐴 ∘ ∅))
63		co02 6264	. . . . . . . 8 ⊢ (𝐴 ∘ ∅) = ∅
64	62, 63	eqtrdi 2814	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ∘ 𝑥) = ∅)
65	64	oveq2d 7428	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐺 Σg (𝐴 ∘ 𝑥)) = (𝐺 Σg ∅))
66		eqid 2763	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
67	66	gsum0 18743	. . . . . 6 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
68	65, 67	eqtrdi 2814	. . . . 5 ⊢ (𝑥 = ∅ → (𝐺 Σg (𝐴 ∘ 𝑥)) = (0g‘𝐺))
69	68, 15, 47	fvmpt3i 6997	. . . 4 ⊢ (∅ ∈ Word 𝐼 → (𝐸‘∅) = (0g‘𝐺))
70	61, 69	mp1i 14	. . 3 ⊢ (𝜑 → (𝐸‘∅) = (0g‘𝐺))
71	21, 60, 70	3jca 1146	. 2 ⊢ (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)) ∧ (𝐸‘∅) = (0g‘𝐺)))
72	2	frmd0 18920	. . 3 ⊢ ∅ = (0g‘𝑀)
73	17, 12, 39, 35, 72, 66	ismhm 18844	. 2 ⊢ (𝐸 ∈ (𝑀 MndHom 𝐺) ↔ ((𝑀 ∈ Mnd ∧ 𝐺 ∈ Mnd) ∧ (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+g‘𝐺)(𝐸‘𝑧)) ∧ (𝐸‘∅) = (0g‘𝐺))))
74	4, 5, 71, 73	syl21anbrc 1363	1 ⊢ (𝜑 → 𝐸 ∈ (𝑀 MndHom 𝐺))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∅c0 4287   ↦ cmpt 5193   ∘ ccom 5667  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  Word cword 14552   ++ cconcat 14609  Basecbs 17270  +_gcplusg 17311  0_gc0g 17493   Σ_g cgsu 17494  Mndcmnd 18793   MndHom cmhm 18840  freeMndcfrmd 18907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-word 14553  df-concat 14610  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-0g 17495  df-gsum 17496  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-submnd 18843  df-frmd 18909

This theorem is referenced by:  frmdup3  18927