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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 | 0ex 5203 | . . . . . . . . . . . 12 ⊢ ∅ ∈ V | |
2 | 0frgp.g | . . . . . . . . . . . . 13 ⊢ 𝐺 = (freeGrp‘∅) | |
3 | 2 | frgpgrp 18882 | . . . . . . . . . . . 12 ⊢ (∅ ∈ V → 𝐺 ∈ Grp) |
4 | 1, 3 | ax-mp 5 | . . . . . . . . . . 11 ⊢ 𝐺 ∈ Grp |
5 | f0 6554 | . . . . . . . . . . 11 ⊢ ∅:∅⟶𝐵 | |
6 | 0frgp.b | . . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺) | |
7 | eqid 2821 | . . . . . . . . . . . . . . . 16 ⊢ ( ~FG ‘∅) = ( ~FG ‘∅) | |
8 | eqid 2821 | . . . . . . . . . . . . . . . 16 ⊢ (varFGrp‘∅) = (varFGrp‘∅) | |
9 | 7, 8, 2, 6 | vrgpf 18888 | . . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (varFGrp‘∅):∅⟶𝐵) |
10 | ffn 6508 | . . . . . . . . . . . . . . 15 ⊢ ((varFGrp‘∅):∅⟶𝐵 → (varFGrp‘∅) Fn ∅) | |
11 | 1, 9, 10 | mp2b 10 | . . . . . . . . . . . . . 14 ⊢ (varFGrp‘∅) Fn ∅ |
12 | fn0 6473 | . . . . . . . . . . . . . 14 ⊢ ((varFGrp‘∅) Fn ∅ ↔ (varFGrp‘∅) = ∅) | |
13 | 11, 12 | mpbi 232 | . . . . . . . . . . . . 13 ⊢ (varFGrp‘∅) = ∅ |
14 | 13 | eqcomi 2830 | . . . . . . . . . . . 12 ⊢ ∅ = (varFGrp‘∅) |
15 | 2, 6, 14 | frgpup3 18898 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ∅ ∈ V ∧ ∅:∅⟶𝐵) → ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅) |
16 | 4, 1, 5, 15 | mp3an 1457 | . . . . . . . . . 10 ⊢ ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅ |
17 | reurmo 3433 | . . . . . . . . . 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 18367 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)) |
20 | 4, 19 | ax-mp 5 | . . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) |
21 | tru 1537 | . . . . . . . . . 10 ⊢ ⊤ | |
22 | 20, 21 | pm3.2i 473 | . . . . . . . . 9 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) |
23 | eqid 2821 | . . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
24 | 23, 6 | 0ghm 18366 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) → (𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺)) |
25 | 4, 4, 24 | mp2an 690 | . . . . . . . . . 10 ⊢ (𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) |
26 | 25, 21 | pm3.2i 473 | . . . . . . . . 9 ⊢ ((𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) |
27 | co02 6107 | . . . . . . . . . . . 12 ⊢ (𝑓 ∘ ∅) = ∅ | |
28 | 27 | bitru 1542 | . . . . . . . . . . 11 ⊢ ((𝑓 ∘ ∅) = ∅ ↔ ⊤) |
29 | 28 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ 𝐵) → ((𝑓 ∘ ∅) = ∅ ↔ ⊤)) |
30 | 28 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑓 = (𝐵 × {(0g‘𝐺)}) → ((𝑓 ∘ ∅) = ∅ ↔ ⊤)) |
31 | 29, 30 | rmoi 3874 | . . . . . . . . 9 ⊢ ((∃*𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅ ∧ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) ∧ ((𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤)) → ( I ↾ 𝐵) = (𝐵 × {(0g‘𝐺)})) |
32 | 18, 22, 26, 31 | mp3an 1457 | . . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝐵 × {(0g‘𝐺)}) |
33 | mptresid 5912 | . . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝑥) | |
34 | fconstmpt 5608 | . . . . . . . 8 ⊢ (𝐵 × {(0g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) | |
35 | 32, 33, 34 | 3eqtr3i 2852 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) |
36 | mpteqb 6781 | . . . . . . . 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 3152 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) ↔ ∀𝑥 ∈ 𝐵 𝑥 = (0g‘𝐺)) |
39 | 35, 38 | mpbi 232 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 𝑥 = (0g‘𝐺) |
40 | 39 | rspec 3207 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0g‘𝐺)) |
41 | velsn 4576 | . . . . 5 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
42 | 40, 41 | sylibr 236 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0g‘𝐺)}) |
43 | 42 | ssriv 3970 | . . 3 ⊢ 𝐵 ⊆ {(0g‘𝐺)} |
44 | 6, 23 | grpidcl 18125 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) |
45 | 4, 44 | ax-mp 5 | . . . 4 ⊢ (0g‘𝐺) ∈ 𝐵 |
46 | snssi 4734 | . . . 4 ⊢ ((0g‘𝐺) ∈ 𝐵 → {(0g‘𝐺)} ⊆ 𝐵) | |
47 | 45, 46 | ax-mp 5 | . . 3 ⊢ {(0g‘𝐺)} ⊆ 𝐵 |
48 | 43, 47 | eqssi 3982 | . 2 ⊢ 𝐵 = {(0g‘𝐺)} |
49 | fvex 6677 | . . 3 ⊢ (0g‘𝐺) ∈ V | |
50 | 49 | ensn1 8567 | . 2 ⊢ {(0g‘𝐺)} ≈ 1o |
51 | 48, 50 | eqbrtri 5079 | 1 ⊢ 𝐵 ≈ 1o |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ∀wral 3138 ∃!wreu 3140 ∃*wrmo 3141 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 {csn 4560 class class class wbr 5058 ↦ cmpt 5138 I cid 5453 × cxp 5547 ↾ cres 5551 ∘ ccom 5553 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 1oc1o 8089 ≈ cen 8500 Basecbs 16477 0gc0g 16707 Grpcgrp 18097 GrpHom cghm 18349 ~FG cefg 18826 freeGrpcfrgp 18827 varFGrpcvrgp 18828 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-splice 14106 df-reverse 14115 df-s2 14204 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-0g 16709 df-gsum 16710 df-imas 16775 df-qus 16776 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-frmd 18008 df-vrmd 18009 df-grp 18100 df-minusg 18101 df-ghm 18350 df-efg 18829 df-frgp 18830 df-vrgp 18831 |
This theorem is referenced by: frgpcyg 20714 |
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