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Mirrors > Home > MPE Home > Th. List > 0frgp | Structured version Visualization version GIF version |
Description: The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
0frgp.g | ⊢ 𝐺 = (freeGrp‘∅) |
0frgp.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
0frgp | ⊢ 𝐵 ≈ 1𝑜 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptresid 5456 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = ( I ↾ 𝐵) | |
2 | 0ex 4790 | . . . . . . . . . . . 12 ⊢ ∅ ∈ V | |
3 | 0frgp.g | . . . . . . . . . . . . 13 ⊢ 𝐺 = (freeGrp‘∅) | |
4 | 3 | frgpgrp 18175 | . . . . . . . . . . . 12 ⊢ (∅ ∈ V → 𝐺 ∈ Grp) |
5 | 2, 4 | ax-mp 5 | . . . . . . . . . . 11 ⊢ 𝐺 ∈ Grp |
6 | f0 6086 | . . . . . . . . . . 11 ⊢ ∅:∅⟶𝐵 | |
7 | 0frgp.b | . . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺) | |
8 | eqid 2622 | . . . . . . . . . . . . . . . 16 ⊢ ( ~FG ‘∅) = ( ~FG ‘∅) | |
9 | eqid 2622 | . . . . . . . . . . . . . . . 16 ⊢ (varFGrp‘∅) = (varFGrp‘∅) | |
10 | 8, 9, 3, 7 | vrgpf 18181 | . . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (varFGrp‘∅):∅⟶𝐵) |
11 | ffn 6045 | . . . . . . . . . . . . . . 15 ⊢ ((varFGrp‘∅):∅⟶𝐵 → (varFGrp‘∅) Fn ∅) | |
12 | 2, 10, 11 | mp2b 10 | . . . . . . . . . . . . . 14 ⊢ (varFGrp‘∅) Fn ∅ |
13 | fn0 6011 | . . . . . . . . . . . . . 14 ⊢ ((varFGrp‘∅) Fn ∅ ↔ (varFGrp‘∅) = ∅) | |
14 | 12, 13 | mpbi 220 | . . . . . . . . . . . . 13 ⊢ (varFGrp‘∅) = ∅ |
15 | 14 | eqcomi 2631 | . . . . . . . . . . . 12 ⊢ ∅ = (varFGrp‘∅) |
16 | 3, 7, 15 | frgpup3 18191 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ∅ ∈ V ∧ ∅:∅⟶𝐵) → ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅) |
17 | 5, 2, 6, 16 | mp3an 1424 | . . . . . . . . . 10 ⊢ ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅ |
18 | reurmo 3161 | . . . . . . . . . 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 17675 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)) |
21 | 5, 20 | ax-mp 5 | . . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) |
22 | tru 1487 | . . . . . . . . . 10 ⊢ ⊤ | |
23 | 21, 22 | pm3.2i 471 | . . . . . . . . 9 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) |
24 | eqid 2622 | . . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
25 | 24, 7 | 0ghm 17674 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) → (𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺)) |
26 | 5, 5, 25 | mp2an 708 | . . . . . . . . . 10 ⊢ (𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) |
27 | 26, 22 | pm3.2i 471 | . . . . . . . . 9 ⊢ ((𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) |
28 | co02 5649 | . . . . . . . . . . . 12 ⊢ (𝑓 ∘ ∅) = ∅ | |
29 | 28 | bitru 1496 | . . . . . . . . . . 11 ⊢ ((𝑓 ∘ ∅) = ∅ ↔ ⊤) |
30 | 29 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ 𝐵) → ((𝑓 ∘ ∅) = ∅ ↔ ⊤)) |
31 | 29 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑓 = (𝐵 × {(0g‘𝐺)}) → ((𝑓 ∘ ∅) = ∅ ↔ ⊤)) |
32 | 30, 31 | rmoi 3530 | . . . . . . . . 9 ⊢ ((∃*𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅ ∧ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) ∧ ((𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)) → ( I ↾ 𝐵) = (𝐵 × {(0g‘𝐺)})) |
33 | 19, 23, 27, 32 | mp3an 1424 | . . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝐵 × {(0g‘𝐺)}) |
34 | fconstmpt 5163 | . . . . . . . 8 ⊢ (𝐵 × {(0g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) | |
35 | 1, 33, 34 | 3eqtri 2648 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) |
36 | mpteqb 6299 | . . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) ↔ ∀𝑥 ∈ 𝐵 𝑥 = (0g‘𝐺))) | |
37 | id 22 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐵) | |
38 | 36, 37 | mprg 2926 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) ↔ ∀𝑥 ∈ 𝐵 𝑥 = (0g‘𝐺)) |
39 | 35, 38 | mpbi 220 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 𝑥 = (0g‘𝐺) |
40 | 39 | rspec 2931 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0g‘𝐺)) |
41 | velsn 4193 | . . . . 5 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
42 | 40, 41 | sylibr 224 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0g‘𝐺)}) |
43 | 42 | ssriv 3607 | . . 3 ⊢ 𝐵 ⊆ {(0g‘𝐺)} |
44 | 7, 24 | grpidcl 17450 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) |
45 | 5, 44 | ax-mp 5 | . . . 4 ⊢ (0g‘𝐺) ∈ 𝐵 |
46 | snssi 4339 | . . . 4 ⊢ ((0g‘𝐺) ∈ 𝐵 → {(0g‘𝐺)} ⊆ 𝐵) | |
47 | 45, 46 | ax-mp 5 | . . 3 ⊢ {(0g‘𝐺)} ⊆ 𝐵 |
48 | 43, 47 | eqssi 3619 | . 2 ⊢ 𝐵 = {(0g‘𝐺)} |
49 | fvex 6201 | . . 3 ⊢ (0g‘𝐺) ∈ V | |
50 | 49 | ensn1 8020 | . 2 ⊢ {(0g‘𝐺)} ≈ 1𝑜 |
51 | 48, 50 | eqbrtri 4674 | 1 ⊢ 𝐵 ≈ 1𝑜 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ∀wral 2912 ∃!wreu 2914 ∃*wrmo 2915 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 ↾ cres 5116 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 ≈ cen 7952 Basecbs 15857 0gc0g 16100 Grpcgrp 17422 GrpHom cghm 17657 ~FG cefg 18119 freeGrpcfrgp 18120 varFGrpcvrgp 18121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-0g 16102 df-gsum 16103 df-imas 16168 df-qus 16169 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-frmd 17386 df-vrmd 17387 df-grp 17425 df-minusg 17426 df-ghm 17658 df-efg 18122 df-frgp 18123 df-vrgp 18124 |
This theorem is referenced by: frgpcyg 19922 |
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