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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 5214 | . . . . . . . . . . . 12 ⊢ ∅ ∈ V | |
2 | 0frgp.g | . . . . . . . . . . . . 13 ⊢ 𝐺 = (freeGrp‘∅) | |
3 | 2 | frgpgrp 18891 | . . . . . . . . . . . 12 ⊢ (∅ ∈ V → 𝐺 ∈ Grp) |
4 | 1, 3 | ax-mp 5 | . . . . . . . . . . 11 ⊢ 𝐺 ∈ Grp |
5 | f0 6563 | . . . . . . . . . . 11 ⊢ ∅:∅⟶𝐵 | |
6 | 0frgp.b | . . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺) | |
7 | eqid 2824 | . . . . . . . . . . . . . . . 16 ⊢ ( ~FG ‘∅) = ( ~FG ‘∅) | |
8 | eqid 2824 | . . . . . . . . . . . . . . . 16 ⊢ (varFGrp‘∅) = (varFGrp‘∅) | |
9 | 7, 8, 2, 6 | vrgpf 18897 | . . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (varFGrp‘∅):∅⟶𝐵) |
10 | ffn 6517 | . . . . . . . . . . . . . . 15 ⊢ ((varFGrp‘∅):∅⟶𝐵 → (varFGrp‘∅) Fn ∅) | |
11 | 1, 9, 10 | mp2b 10 | . . . . . . . . . . . . . 14 ⊢ (varFGrp‘∅) Fn ∅ |
12 | fn0 6482 | . . . . . . . . . . . . . 14 ⊢ ((varFGrp‘∅) Fn ∅ ↔ (varFGrp‘∅) = ∅) | |
13 | 11, 12 | mpbi 232 | . . . . . . . . . . . . 13 ⊢ (varFGrp‘∅) = ∅ |
14 | 13 | eqcomi 2833 | . . . . . . . . . . . 12 ⊢ ∅ = (varFGrp‘∅) |
15 | 2, 6, 14 | frgpup3 18907 | . . . . . . . . . . 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 3436 | . . . . . . . . . 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 18376 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)) |
20 | 4, 19 | ax-mp 5 | . . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) |
21 | tru 1540 | . . . . . . . . . 10 ⊢ ⊤ | |
22 | 20, 21 | pm3.2i 473 | . . . . . . . . 9 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) |
23 | eqid 2824 | . . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
24 | 23, 6 | 0ghm 18375 | . . . . . . . . . . 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 6116 | . . . . . . . . . . . 12 ⊢ (𝑓 ∘ ∅) = ∅ | |
28 | 27 | bitru 1545 | . . . . . . . . . . 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 3878 | . . . . . . . . 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 5921 | . . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝑥) | |
34 | fconstmpt 5617 | . . . . . . . 8 ⊢ (𝐵 × {(0g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) | |
35 | 32, 33, 34 | 3eqtr3i 2855 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) |
36 | mpteqb 6790 | . . . . . . . 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 3155 | . . . . . . 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 3210 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0g‘𝐺)) |
41 | velsn 4586 | . . . . 5 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
42 | 40, 41 | sylibr 236 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0g‘𝐺)}) |
43 | 42 | ssriv 3974 | . . 3 ⊢ 𝐵 ⊆ {(0g‘𝐺)} |
44 | 6, 23 | grpidcl 18134 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) |
45 | 4, 44 | ax-mp 5 | . . . 4 ⊢ (0g‘𝐺) ∈ 𝐵 |
46 | snssi 4744 | . . . 4 ⊢ ((0g‘𝐺) ∈ 𝐵 → {(0g‘𝐺)} ⊆ 𝐵) | |
47 | 45, 46 | ax-mp 5 | . . 3 ⊢ {(0g‘𝐺)} ⊆ 𝐵 |
48 | 43, 47 | eqssi 3986 | . 2 ⊢ 𝐵 = {(0g‘𝐺)} |
49 | fvex 6686 | . . 3 ⊢ (0g‘𝐺) ∈ V | |
50 | 49 | ensn1 8576 | . 2 ⊢ {(0g‘𝐺)} ≈ 1o |
51 | 48, 50 | eqbrtri 5090 | 1 ⊢ 𝐵 ≈ 1o |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 ∀wral 3141 ∃!wreu 3143 ∃*wrmo 3144 Vcvv 3497 ⊆ wss 3939 ∅c0 4294 {csn 4570 class class class wbr 5069 ↦ cmpt 5149 I cid 5462 × cxp 5556 ↾ cres 5560 ∘ ccom 5562 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 1oc1o 8098 ≈ cen 8509 Basecbs 16486 0gc0g 16716 Grpcgrp 18106 GrpHom cghm 18358 ~FG cefg 18835 freeGrpcfrgp 18836 varFGrpcvrgp 18837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-ot 4579 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-ec 8294 df-qs 8298 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-word 13865 df-lsw 13918 df-concat 13926 df-s1 13953 df-substr 14006 df-pfx 14036 df-splice 14115 df-reverse 14124 df-s2 14213 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-0g 16718 df-gsum 16719 df-imas 16784 df-qus 16785 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-frmd 18017 df-vrmd 18018 df-grp 18109 df-minusg 18110 df-ghm 18359 df-efg 18838 df-frgp 18839 df-vrgp 18840 |
This theorem is referenced by: frgpcyg 20723 |
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