0frgp - Metamath Proof Explorer

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Theorem 0frgp 19812

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 5313	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
2		0frgp.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (freeGrp‘∅)
3	2	frgpgrp 19795	. . . . . . . . . . . 12 ⊢ (∅ ∈ V → 𝐺 ∈ Grp)
4	1, 3	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐺 ∈ Grp
5		f0 6790	. . . . . . . . . . 11 ⊢ ∅:∅⟶𝐵
6		0frgp.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
7		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ ( ~_FG ‘∅) = ( ~_FG ‘∅)
8		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (var_FGrp‘∅) = (var_FGrp‘∅)
9	7, 8, 2, 6	vrgpf 19801	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (var_FGrp‘∅):∅⟶𝐵)
10		ffn 6737	. . . . . . . . . . . . . . 15 ⊢ ((var_FGrp‘∅):∅⟶𝐵 → (var_FGrp‘∅) Fn ∅)
11	1, 9, 10	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (var_FGrp‘∅) Fn ∅
12		fn0 6700	. . . . . . . . . . . . . 14 ⊢ ((var_FGrp‘∅) Fn ∅ ↔ (var_FGrp‘∅) = ∅)
13	11, 12	mpbi 230	. . . . . . . . . . . . 13 ⊢ (var_FGrp‘∅) = ∅
14	13	eqcomi 2744	. . . . . . . . . . . 12 ⊢ ∅ = (var_FGrp‘∅)
15	2, 6, 14	frgpup3 19811	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ∅ ∈ V ∧ ∅:∅⟶𝐵) → ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅)
16	4, 1, 5, 15	mp3an 1460	. . . . . . . . . 10 ⊢ ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅
17		reurmo 3381	. . . . . . . . . 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 19262	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
20	4, 19	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)
21		tru 1541	. . . . . . . . . 10 ⊢ ⊤
22	20, 21	pm3.2i 470	. . . . . . . . 9 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)
23		eqid 2735	. . . . . . . . . . . 12 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
24	23, 6	0ghm 19261	. . . . . . . . . . 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 6282	. . . . . . . . . . . 12 ⊢ (𝑓 ∘ ∅) = ∅
28	27	bitru 1546	. . . . . . . . . . 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 3900	. . . . . . . . 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 1460	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝐵 × {(0_g‘𝐺)})
33		mptresid 6071	. . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝑥)
34		fconstmpt 5751	. . . . . . . 8 ⊢ (𝐵 × {(0_g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
35	32, 33, 34	3eqtr3i 2771	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0_g‘𝐺))
36		mpteqb 7035	. . . . . . . 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 3065	. . . . . . 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 3248	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0_g‘𝐺))
41		velsn 4647	. . . . 5 ⊢ (𝑥 ∈ {(0_g‘𝐺)} ↔ 𝑥 = (0_g‘𝐺))
42	40, 41	sylibr 234	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0_g‘𝐺)})
43	42	ssriv 3999	. . 3 ⊢ 𝐵 ⊆ {(0_g‘𝐺)}
44	6, 23	grpidcl 18996	. . . . 5 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝐵)
45	4, 44	ax-mp 5	. . . 4 ⊢ (0_g‘𝐺) ∈ 𝐵
46		snssi 4813	. . . 4 ⊢ ((0_g‘𝐺) ∈ 𝐵 → {(0_g‘𝐺)} ⊆ 𝐵)
47	45, 46	ax-mp 5	. . 3 ⊢ {(0_g‘𝐺)} ⊆ 𝐵
48	43, 47	eqssi 4012	. 2 ⊢ 𝐵 = {(0_g‘𝐺)}
49		fvex 6920	. . 3 ⊢ (0_g‘𝐺) ∈ V
50	49	ensn1 9060	. 2 ⊢ {(0_g‘𝐺)} ≈ 1_o
51	48, 50	eqbrtri 5169	1 ⊢ 𝐵 ≈ 1_o

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ∀wral 3059 ∃!wreu 3376 ∃*wrmo 3377 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 {csn 4631 class class class wbr 5148 ↦ cmpt 5231 I cid 5582 × cxp 5687 ↾ cres 5691 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1_oc1o 8498 ≈ cen 8981 Basecbs 17245 0_gc0g 17486 Grpcgrp 18964 GrpHom cghm 19243 ~_FG cefg 19739 freeGrpcfrgp 19740 var_FGrpcvrgp 19741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-reverse 14794 df-s2 14884 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17488 df-gsum 17489 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-frmd 18875 df-vrmd 18876 df-grp 18967 df-minusg 18968 df-ghm 19244 df-efg 19742 df-frgp 19743 df-vrgp 19744

This theorem is referenced by: frgpcyg 21610

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