0frgp - Metamath Proof Explorer

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Theorem 0frgp 18589

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 5714	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = ( I ↾ 𝐵)
2		0ex 5028	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
3		0frgp.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (freeGrp‘∅)
4	3	frgpgrp 18572	. . . . . . . . . . . 12 ⊢ (∅ ∈ V → 𝐺 ∈ Grp)
5	2, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐺 ∈ Grp
6		f0 6338	. . . . . . . . . . 11 ⊢ ∅:∅⟶𝐵
7		0frgp.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
8		eqid 2778	. . . . . . . . . . . . . . . 16 ⊢ ( ~_FG ‘∅) = ( ~_FG ‘∅)
9		eqid 2778	. . . . . . . . . . . . . . . 16 ⊢ (var_FGrp‘∅) = (var_FGrp‘∅)
10	8, 9, 3, 7	vrgpf 18578	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (var_FGrp‘∅):∅⟶𝐵)
11		ffn 6293	. . . . . . . . . . . . . . 15 ⊢ ((var_FGrp‘∅):∅⟶𝐵 → (var_FGrp‘∅) Fn ∅)
12	2, 10, 11	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (var_FGrp‘∅) Fn ∅
13		fn0 6259	. . . . . . . . . . . . . 14 ⊢ ((var_FGrp‘∅) Fn ∅ ↔ (var_FGrp‘∅) = ∅)
14	12, 13	mpbi 222	. . . . . . . . . . . . 13 ⊢ (var_FGrp‘∅) = ∅
15	14	eqcomi 2787	. . . . . . . . . . . 12 ⊢ ∅ = (var_FGrp‘∅)
16	3, 7, 15	frgpup3 18588	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ∅ ∈ V ∧ ∅:∅⟶𝐵) → ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅)
17	5, 2, 6, 16	mp3an 1534	. . . . . . . . . 10 ⊢ ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅
18		reurmo 3357	. . . . . . . . . 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 18070	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
21	5, 20	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)
22		tru 1606	. . . . . . . . . 10 ⊢ ⊤
23	21, 22	pm3.2i 464	. . . . . . . . 9 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)
24		eqid 2778	. . . . . . . . . . . 12 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
25	24, 7	0ghm 18069	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) → (𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺))
26	5, 5, 25	mp2an 682	. . . . . . . . . 10 ⊢ (𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺)
27	26, 22	pm3.2i 464	. . . . . . . . 9 ⊢ ((𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)
28		co02 5905	. . . . . . . . . . . 12 ⊢ (𝑓 ∘ ∅) = ∅
29	28	bitru 1611	. . . . . . . . . . 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 3748	. . . . . . . . 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 1534	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝐵 × {(0_g‘𝐺)})
34		fconstmpt 5413	. . . . . . . 8 ⊢ (𝐵 × {(0_g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
35	1, 33, 34	3eqtri 2806	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
36		mpteqb 6562	. . . . . . . 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 3108	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺)) ↔ ∀𝑥 ∈ 𝐵 𝑥 = (0_g‘𝐺))
39	35, 38	mpbi 222	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 𝑥 = (0_g‘𝐺)
40	39	rspec 3113	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0_g‘𝐺))
41		velsn 4414	. . . . 5 ⊢ (𝑥 ∈ {(0_g‘𝐺)} ↔ 𝑥 = (0_g‘𝐺))
42	40, 41	sylibr 226	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0_g‘𝐺)})
43	42	ssriv 3825	. . 3 ⊢ 𝐵 ⊆ {(0_g‘𝐺)}
44	7, 24	grpidcl 17848	. . . . 5 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝐵)
45	5, 44	ax-mp 5	. . . 4 ⊢ (0_g‘𝐺) ∈ 𝐵
46		snssi 4572	. . . 4 ⊢ ((0_g‘𝐺) ∈ 𝐵 → {(0_g‘𝐺)} ⊆ 𝐵)
47	45, 46	ax-mp 5	. . 3 ⊢ {(0_g‘𝐺)} ⊆ 𝐵
48	43, 47	eqssi 3837	. 2 ⊢ 𝐵 = {(0_g‘𝐺)}
49		fvex 6461	. . 3 ⊢ (0_g‘𝐺) ∈ V
50	49	ensn1 8307	. 2 ⊢ {(0_g‘𝐺)} ≈ 1_o
51	48, 50	eqbrtri 4909	1 ⊢ 𝐵 ≈ 1_o

Colors of variables: wff setvar class

Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ⊤wtru 1602 ∈ wcel 2107 ∀wral 3090 ∃!wreu 3092 ∃*wrmo 3093 Vcvv 3398 ⊆ wss 3792 ∅c0 4141 {csn 4398 class class class wbr 4888 ↦ cmpt 4967 I cid 5262 × cxp 5355 ↾ cres 5359 ∘ ccom 5361 Fn wfn 6132 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 1_oc1o 7838 ≈ cen 8240 Basecbs 16266 0_gc0g 16497 Grpcgrp 17820 GrpHom cghm 18052 ~_FG cefg 18514 freeGrpcfrgp 18515 var_FGrpcvrgp 18516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-ec 8030 df-qs 8034 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-inf 8639 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-xnn0 11720 df-z 11734 df-dec 11851 df-uz 11998 df-fz 12649 df-fzo 12790 df-seq 13125 df-hash 13442 df-word 13606 df-lsw 13659 df-concat 13667 df-s1 13692 df-substr 13737 df-pfx 13786 df-splice 13893 df-reverse 13911 df-s2 14005 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-0g 16499 df-gsum 16500 df-imas 16565 df-qus 16566 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mhm 17732 df-submnd 17733 df-frmd 17784 df-vrmd 17785 df-grp 17823 df-minusg 17824 df-ghm 18053 df-efg 18517 df-frgp 18518 df-vrgp 18519

This theorem is referenced by: frgpcyg 20328

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