0frgp - Metamath Proof Explorer

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Theorem 0frgp 18392

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		mptresid 5614	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = ( I ↾ 𝐵)
2		0ex 4942	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
3		0frgp.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (freeGrp‘∅)
4	3	frgpgrp 18375	. . . . . . . . . . . 12 ⊢ (∅ ∈ V → 𝐺 ∈ Grp)
5	2, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐺 ∈ Grp
6		f0 6247	. . . . . . . . . . 11 ⊢ ∅:∅⟶𝐵
7		0frgp.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
8		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ ( ~_FG ‘∅) = ( ~_FG ‘∅)
9		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (var_FGrp‘∅) = (var_FGrp‘∅)
10	8, 9, 3, 7	vrgpf 18381	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (var_FGrp‘∅):∅⟶𝐵)
11		ffn 6206	. . . . . . . . . . . . . . 15 ⊢ ((var_FGrp‘∅):∅⟶𝐵 → (var_FGrp‘∅) Fn ∅)
12	2, 10, 11	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (var_FGrp‘∅) Fn ∅
13		fn0 6172	. . . . . . . . . . . . . 14 ⊢ ((var_FGrp‘∅) Fn ∅ ↔ (var_FGrp‘∅) = ∅)
14	12, 13	mpbi 220	. . . . . . . . . . . . 13 ⊢ (var_FGrp‘∅) = ∅
15	14	eqcomi 2769	. . . . . . . . . . . 12 ⊢ ∅ = (var_FGrp‘∅)
16	3, 7, 15	frgpup3 18391	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ∅ ∈ V ∧ ∅:∅⟶𝐵) → ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅)
17	5, 2, 6, 16	mp3an 1573	. . . . . . . . . 10 ⊢ ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅
18		reurmo 3300	. . . . . . . . . 10 ⊢ (∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅ → ∃*𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅)
19	17, 18	ax-mp 5	. . . . . . . . 9 ⊢ ∃*𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅
20	7	idghm 17876	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
21	5, 20	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)
22		tru 1636	. . . . . . . . . 10 ⊢ ⊤
23	21, 22	pm3.2i 470	. . . . . . . . 9 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)
24		eqid 2760	. . . . . . . . . . . 12 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
25	24, 7	0ghm 17875	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) → (𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺))
26	5, 5, 25	mp2an 710	. . . . . . . . . 10 ⊢ (𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺)
27	26, 22	pm3.2i 470	. . . . . . . . 9 ⊢ ((𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)
28		co02 5810	. . . . . . . . . . . 12 ⊢ (𝑓 ∘ ∅) = ∅
29	28	bitru 1645	. . . . . . . . . . 11 ⊢ ((𝑓 ∘ ∅) = ∅ ↔ ⊤)
30	29	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ 𝐵) → ((𝑓 ∘ ∅) = ∅ ↔ ⊤))
31	29	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 = (𝐵 × {(0_g‘𝐺)}) → ((𝑓 ∘ ∅) = ∅ ↔ ⊤))
32	30, 31	rmoi 3671	. . . . . . . . 9 ⊢ ((∃*𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅ ∧ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) ∧ ((𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)) → ( I ↾ 𝐵) = (𝐵 × {(0_g‘𝐺)}))
33	19, 23, 27, 32	mp3an 1573	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝐵 × {(0_g‘𝐺)})
34		fconstmpt 5320	. . . . . . . 8 ⊢ (𝐵 × {(0_g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
35	1, 33, 34	3eqtri 2786	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
36		mpteqb 6461	. . . . . . . 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 3064	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺)) ↔ ∀𝑥 ∈ 𝐵 𝑥 = (0_g‘𝐺))
39	35, 38	mpbi 220	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 𝑥 = (0_g‘𝐺)
40	39	rspec 3069	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0_g‘𝐺))
41		velsn 4337	. . . . 5 ⊢ (𝑥 ∈ {(0_g‘𝐺)} ↔ 𝑥 = (0_g‘𝐺))
42	40, 41	sylibr 224	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0_g‘𝐺)})
43	42	ssriv 3748	. . 3 ⊢ 𝐵 ⊆ {(0_g‘𝐺)}
44	7, 24	grpidcl 17651	. . . . 5 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝐵)
45	5, 44	ax-mp 5	. . . 4 ⊢ (0_g‘𝐺) ∈ 𝐵
46		snssi 4484	. . . 4 ⊢ ((0_g‘𝐺) ∈ 𝐵 → {(0_g‘𝐺)} ⊆ 𝐵)
47	45, 46	ax-mp 5	. . 3 ⊢ {(0_g‘𝐺)} ⊆ 𝐵
48	43, 47	eqssi 3760	. 2 ⊢ 𝐵 = {(0_g‘𝐺)}
49		fvex 6362	. . 3 ⊢ (0_g‘𝐺) ∈ V
50	49	ensn1 8185	. 2 ⊢ {(0_g‘𝐺)} ≈ 1_𝑜
51	48, 50	eqbrtri 4825	1 ⊢ 𝐵 ≈ 1_𝑜

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1632 ⊤wtru 1633 ∈ wcel 2139 ∀wral 3050 ∃!wreu 3052 ∃*wrmo 3053 Vcvv 3340 ⊆ wss 3715 ∅c0 4058 {csn 4321 class class class wbr 4804 ↦ cmpt 4881 I cid 5173 × cxp 5264 ↾ cres 5268 ∘ ccom 5270 Fn wfn 6044 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 1_𝑜c1o 7722 ≈ cen 8118 Basecbs 16059 0_gc0g 16302 Grpcgrp 17623 GrpHom cghm 17858 ~_FG cefg 18319 freeGrpcfrgp 18320 var_FGrpcvrgp 18321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-ot 4330 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-ec 7913 df-qs 7917 df-map 8025 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-inf 8514 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-xnn0 11556 df-z 11570 df-dec 11686 df-uz 11880 df-fz 12520 df-fzo 12660 df-seq 12996 df-hash 13312 df-word 13485 df-lsw 13486 df-concat 13487 df-s1 13488 df-substr 13489 df-splice 13490 df-reverse 13491 df-s2 13793 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-0g 16304 df-gsum 16305 df-imas 16370 df-qus 16371 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-submnd 17537 df-frmd 17587 df-vrmd 17588 df-grp 17626 df-minusg 17627 df-ghm 17859 df-efg 18322 df-frgp 18323 df-vrgp 18324

This theorem is referenced by: frgpcyg 20124

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