0frgp - Metamath Proof Explorer

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Theorem 0frgp 19743

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 5242	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
2		0frgp.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (freeGrp‘∅)
3	2	frgpgrp 19726	. . . . . . . . . . . 12 ⊢ (∅ ∈ V → 𝐺 ∈ Grp)
4	1, 3	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐺 ∈ Grp
5		f0 6713	. . . . . . . . . . 11 ⊢ ∅:∅⟶𝐵
6		0frgp.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
7		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ ( ~_FG ‘∅) = ( ~_FG ‘∅)
8		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (var_FGrp‘∅) = (var_FGrp‘∅)
9	7, 8, 2, 6	vrgpf 19732	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (var_FGrp‘∅):∅⟶𝐵)
10		ffn 6660	. . . . . . . . . . . . . . 15 ⊢ ((var_FGrp‘∅):∅⟶𝐵 → (var_FGrp‘∅) Fn ∅)
11	1, 9, 10	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (var_FGrp‘∅) Fn ∅
12		fn0 6621	. . . . . . . . . . . . . 14 ⊢ ((var_FGrp‘∅) Fn ∅ ↔ (var_FGrp‘∅) = ∅)
13	11, 12	mpbi 230	. . . . . . . . . . . . 13 ⊢ (var_FGrp‘∅) = ∅
14	13	eqcomi 2746	. . . . . . . . . . . 12 ⊢ ∅ = (var_FGrp‘∅)
15	2, 6, 14	frgpup3 19742	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ∅ ∈ V ∧ ∅:∅⟶𝐵) → ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅)
16	4, 1, 5, 15	mp3an 1464	. . . . . . . . . 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 19195	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
20	4, 19	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)
21		tru 1546	. . . . . . . . . 10 ⊢ ⊤
22	20, 21	pm3.2i 470	. . . . . . . . 9 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)
23		eqid 2737	. . . . . . . . . . . 12 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
24	23, 6	0ghm 19194	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) → (𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺))
25	4, 4, 24	mp2an 693	. . . . . . . . . 10 ⊢ (𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺)
26	25, 21	pm3.2i 470	. . . . . . . . 9 ⊢ ((𝐵 × {(0_g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)
27		co02 6217	. . . . . . . . . . . 12 ⊢ (𝑓 ∘ ∅) = ∅
28	27	bitru 1551	. . . . . . . . . . 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 3830	. . . . . . . . 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 1464	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝐵 × {(0_g‘𝐺)})
33		mptresid 6008	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝑥)
34		fconstmpt 5684	. . . . . . . 8 ⊢ (𝐵 × {(0_g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
35	32, 33, 34	3eqtr3i 2768	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
36		mpteqb 6959	. . . . . . . 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 3058	. . . . . . 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 3229	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0_g‘𝐺))
41		velsn 4584	. . . . 5 ⊢ (𝑥 ∈ {(0_g‘𝐺)} ↔ 𝑥 = (0_g‘𝐺))
42	40, 41	sylibr 234	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0_g‘𝐺)})
43	42	ssriv 3926	. . 3 ⊢ 𝐵 ⊆ {(0_g‘𝐺)}
44	6, 23	grpidcl 18930	. . . . 5 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝐵)
45	4, 44	ax-mp 5	. . . 4 ⊢ (0_g‘𝐺) ∈ 𝐵
46		snssi 4752	. . . 4 ⊢ ((0_g‘𝐺) ∈ 𝐵 → {(0_g‘𝐺)} ⊆ 𝐵)
47	45, 46	ax-mp 5	. . 3 ⊢ {(0_g‘𝐺)} ⊆ 𝐵
48	43, 47	eqssi 3939	. 2 ⊢ 𝐵 = {(0_g‘𝐺)}
49		fvex 6845	. . 3 ⊢ (0_g‘𝐺) ∈ V
50	49	ensn1 8959	. 2 ⊢ {(0_g‘𝐺)} ≈ 1_o
51	48, 50	eqbrtri 5107	1 ⊢ 𝐵 ≈ 1_o

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ∀wral 3052 ∃!wreu 3341 ∃*wrmo 3342 Vcvv 3430 ⊆ wss 3890 ∅c0 4274 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 I cid 5516 × cxp 5620 ↾ cres 5624 ∘ ccom 5626 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 1_oc1o 8389 ≈ cen 8881 Basecbs 17168 0_gc0g 17391 Grpcgrp 18898 GrpHom cghm 19176 ~_FG cefg 19670 freeGrpcfrgp 19671 var_FGrpcvrgp 19672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-ec 8636 df-qs 8640 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9346 df-inf 9347 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-xnn0 12500 df-z 12514 df-dec 12634 df-uz 12778 df-fz 13451 df-fzo 13598 df-seq 13953 df-hash 14282 df-word 14465 df-lsw 14514 df-concat 14522 df-s1 14548 df-substr 14593 df-pfx 14623 df-splice 14701 df-reverse 14710 df-s2 14799 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-0g 17393 df-gsum 17394 df-imas 17461 df-qus 17462 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-frmd 18806 df-vrmd 18807 df-grp 18901 df-minusg 18902 df-ghm 19177 df-efg 19673 df-frgp 19674 df-vrgp 19675

This theorem is referenced by: frgpcyg 21561

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