0frgp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 0frgp		Structured version Visualization version GIF version

Theorem 0frgp 19658

Description: The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Hypotheses**
Ref	Expression
0frgp.g	⊢ 𝐺 = (freeGrp‘∅)
0frgp.b	⊢ 𝐵 = (Base‘𝐺)

**Assertion**
Ref	Expression
0frgp	⊢ 𝐵 ≈ 1_o

Proof of Theorem 0frgp Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5246	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
2		0frgp.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (freeGrp‘∅)
3	2	frgpgrp 19641	. . . . . . . . . . . 12 ⊢ (∅ ∈ V → 𝐺 ∈ Grp)
4	1, 3	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐺 ∈ Grp
5		f0 6705	. . . . . . . . . . 11 ⊢ ∅:∅⟶𝐵
6		0frgp.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
7		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ ( ~_FG ‘∅) = ( ~_FG ‘∅)
8		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (var_FGrp‘∅) = (var_FGrp‘∅)
9	7, 8, 2, 6	vrgpf 19647	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (var_FGrp‘∅):∅⟶𝐵)
10		ffn 6652	. . . . . . . . . . . . . . 15 ⊢ ((var_FGrp‘∅):∅⟶𝐵 → (var_FGrp‘∅) Fn ∅)
11	1, 9, 10	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (var_FGrp‘∅) Fn ∅
12		fn0 6613	. . . . . . . . . . . . . 14 ⊢ ((var_FGrp‘∅) Fn ∅ ↔ (var_FGrp‘∅) = ∅)
13	11, 12	mpbi 230	. . . . . . . . . . . . 13 ⊢ (var_FGrp‘∅) = ∅
14	13	eqcomi 2738	. . . . . . . . . . . 12 ⊢ ∅ = (var_FGrp‘∅)
15	2, 6, 14	frgpup3 19657	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ∅ ∈ V ∧ ∅:∅⟶𝐵) → ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅)
16	4, 1, 5, 15	mp3an 1463	. . . . . . . . . 10 ⊢ ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅
17		reurmo 3346	. . . . . . . . . 10 ⊢ (∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅ → ∃*𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅)
18	16, 17	ax-mp 5	. . . . . . . . 9 ⊢ ∃*𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅
19	6	idghm 19110	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
20	4, 19	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)
21		tru 1544	. . . . . . . . . 10 ⊢ ⊤
22	20, 21	pm3.2i 470	. . . . . . . . 9 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)
23		eqid 2729	. . . . . . . . . . . 12 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
24	23, 6	0ghm 19109	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) → (𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺))
25	4, 4, 24	mp2an 692	. . . . . . . . . 10 ⊢ (𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺)
26	25, 21	pm3.2i 470	. . . . . . . . 9 ⊢ ((𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)
27		co02 6209	. . . . . . . . . . . 12 ⊢ (𝑓 ∘ ∅) = ∅
28	27	bitru 1549	. . . . . . . . . . 11 ⊢ ((𝑓 ∘ ∅) = ∅ ↔ ⊤)
29	28	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ 𝐵) → ((𝑓 ∘ ∅) = ∅ ↔ ⊤))
30	28	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 = (𝐵 × {(0_g‘𝐺)}) → ((𝑓 ∘ ∅) = ∅ ↔ ⊤))
31	29, 30	rmoi 3843	. . . . . . . . 9 ⊢ ((∃*𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅ ∧ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) ∧ ((𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)) → ( I ↾ 𝐵) = (𝐵 × {(0_g‘𝐺)}))
32	18, 22, 26, 31	mp3an 1463	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝐵 × {(0_g‘𝐺)})
33		mptresid 6002	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝑥)
34		fconstmpt 5681	. . . . . . . 8 ⊢ (𝐵 × {(0_g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
35	32, 33, 34	3eqtr3i 2760	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
36		mpteqb 6949	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺)) ↔ ∀𝑥 ∈ 𝐵 𝑥 = (0_g‘𝐺)))
37		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐵)
38	36, 37	mprg 3050	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺)) ↔ ∀𝑥 ∈ 𝐵 𝑥 = (0_g‘𝐺))
39	35, 38	mpbi 230	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 𝑥 = (0_g‘𝐺)
40	39	rspec 3220	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0_g‘𝐺))
41		velsn 4593	. . . . 5 ⊢ (𝑥 ∈ {(0_g‘𝐺)} ↔ 𝑥 = (0_g‘𝐺))
42	40, 41	sylibr 234	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0_g‘𝐺)})
43	42	ssriv 3939	. . 3 ⊢ 𝐵 ⊆ {(0_g‘𝐺)}
44	6, 23	grpidcl 18844	. . . . 5 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝐵)
45	4, 44	ax-mp 5	. . . 4 ⊢ (0_g‘𝐺) ∈ 𝐵
46		snssi 4759	. . . 4 ⊢ ((0_g‘𝐺) ∈ 𝐵 → {(0_g‘𝐺)} ⊆ 𝐵)
47	45, 46	ax-mp 5	. . 3 ⊢ {(0_g‘𝐺)} ⊆ 𝐵
48	43, 47	eqssi 3952	. 2 ⊢ 𝐵 = {(0_g‘𝐺)}
49		fvex 6835	. . 3 ⊢ (0_g‘𝐺) ∈ V
50	49	ensn1 8946	. 2 ⊢ {(0_g‘𝐺)} ≈ 1_o
51	48, 50	eqbrtri 5113	1 ⊢ 𝐵 ≈ 1_o

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ∀wral 3044 ∃!wreu 3341 ∃*wrmo 3342 Vcvv 3436 ⊆ wss 3903 ∅c0 4284 {csn 4577 class class class wbr 5092 ↦ cmpt 5173 I cid 5513 × cxp 5617 ↾ cres 5621 ∘ ccom 5623 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 1_oc1o 8381 ≈ cen 8869 Basecbs 17120 0_gc0g 17343 Grpcgrp 18812 GrpHom cghm 19091 ~_FG cefg 19585 freeGrpcfrgp 19586 var_FGrpcvrgp 19587

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-ec 8627 df-qs 8631 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-xnn0 12458 df-z 12472 df-dec 12592 df-uz 12736 df-fz 13411 df-fzo 13558 df-seq 13909 df-hash 14238 df-word 14421 df-lsw 14470 df-concat 14478 df-s1 14503 df-substr 14548 df-pfx 14578 df-splice 14656 df-reverse 14665 df-s2 14755 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-0g 17345 df-gsum 17346 df-imas 17412 df-qus 17413 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-submnd 18658 df-frmd 18723 df-vrmd 18724 df-grp 18815 df-minusg 18816 df-ghm 19092 df-efg 19588 df-frgp 19589 df-vrgp 19590

This theorem is referenced by: frgpcyg 21480

W3C validator