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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 5306 | . . . . . . . . . . . 12 ⊢ ∅ ∈ V | |
| 2 | 0frgp.g | . . . . . . . . . . . . 13 ⊢ 𝐺 = (freeGrp‘∅) | |
| 3 | 2 | frgpgrp 19781 | . . . . . . . . . . . 12 ⊢ (∅ ∈ V → 𝐺 ∈ Grp) | 
| 4 | 1, 3 | ax-mp 5 | . . . . . . . . . . 11 ⊢ 𝐺 ∈ Grp | 
| 5 | f0 6788 | . . . . . . . . . . 11 ⊢ ∅:∅⟶𝐵 | |
| 6 | 0frgp.b | . . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺) | |
| 7 | eqid 2736 | . . . . . . . . . . . . . . . 16 ⊢ ( ~FG ‘∅) = ( ~FG ‘∅) | |
| 8 | eqid 2736 | . . . . . . . . . . . . . . . 16 ⊢ (varFGrp‘∅) = (varFGrp‘∅) | |
| 9 | 7, 8, 2, 6 | vrgpf 19787 | . . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (varFGrp‘∅):∅⟶𝐵) | 
| 10 | ffn 6735 | . . . . . . . . . . . . . . 15 ⊢ ((varFGrp‘∅):∅⟶𝐵 → (varFGrp‘∅) Fn ∅) | |
| 11 | 1, 9, 10 | mp2b 10 | . . . . . . . . . . . . . 14 ⊢ (varFGrp‘∅) Fn ∅ | 
| 12 | fn0 6698 | . . . . . . . . . . . . . 14 ⊢ ((varFGrp‘∅) Fn ∅ ↔ (varFGrp‘∅) = ∅) | |
| 13 | 11, 12 | mpbi 230 | . . . . . . . . . . . . 13 ⊢ (varFGrp‘∅) = ∅ | 
| 14 | 13 | eqcomi 2745 | . . . . . . . . . . . 12 ⊢ ∅ = (varFGrp‘∅) | 
| 15 | 2, 6, 14 | frgpup3 19797 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ∅ ∈ V ∧ ∅:∅⟶𝐵) → ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅) | 
| 16 | 4, 1, 5, 15 | mp3an 1462 | . . . . . . . . . 10 ⊢ ∃!𝑓 ∈ (𝐺 GrpHom 𝐺)(𝑓 ∘ ∅) = ∅ | 
| 17 | reurmo 3382 | . . . . . . . . . 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 19250 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)) | 
| 20 | 4, 19 | ax-mp 5 | . . . . . . . . . 10 ⊢ ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) | 
| 21 | tru 1543 | . . . . . . . . . 10 ⊢ ⊤ | |
| 22 | 20, 21 | pm3.2i 470 | . . . . . . . . 9 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) | 
| 23 | eqid 2736 | . . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 24 | 23, 6 | 0ghm 19249 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) → (𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺)) | 
| 25 | 4, 4, 24 | mp2an 692 | . . . . . . . . . 10 ⊢ (𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) | 
| 26 | 25, 21 | pm3.2i 470 | . . . . . . . . 9 ⊢ ((𝐵 × {(0g‘𝐺)}) ∈ (𝐺 GrpHom 𝐺) ∧ ⊤) | 
| 27 | co02 6279 | . . . . . . . . . . . 12 ⊢ (𝑓 ∘ ∅) = ∅ | |
| 28 | 27 | bitru 1548 | . . . . . . . . . . 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 3890 | . . . . . . . . 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 1462 | . . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝐵 × {(0g‘𝐺)}) | 
| 33 | mptresid 6068 | . . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝑥) | |
| 34 | fconstmpt 5746 | . . . . . . . 8 ⊢ (𝐵 × {(0g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) | |
| 35 | 32, 33, 34 | 3eqtr3i 2772 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) | 
| 36 | mpteqb 7034 | . . . . . . . 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 3066 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) ↔ ∀𝑥 ∈ 𝐵 𝑥 = (0g‘𝐺)) | 
| 39 | 35, 38 | mpbi 230 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 𝑥 = (0g‘𝐺) | 
| 40 | 39 | rspec 3249 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0g‘𝐺)) | 
| 41 | velsn 4641 | . . . . 5 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
| 42 | 40, 41 | sylibr 234 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0g‘𝐺)}) | 
| 43 | 42 | ssriv 3986 | . . 3 ⊢ 𝐵 ⊆ {(0g‘𝐺)} | 
| 44 | 6, 23 | grpidcl 18984 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) | 
| 45 | 4, 44 | ax-mp 5 | . . . 4 ⊢ (0g‘𝐺) ∈ 𝐵 | 
| 46 | snssi 4807 | . . . 4 ⊢ ((0g‘𝐺) ∈ 𝐵 → {(0g‘𝐺)} ⊆ 𝐵) | |
| 47 | 45, 46 | ax-mp 5 | . . 3 ⊢ {(0g‘𝐺)} ⊆ 𝐵 | 
| 48 | 43, 47 | eqssi 3999 | . 2 ⊢ 𝐵 = {(0g‘𝐺)} | 
| 49 | fvex 6918 | . . 3 ⊢ (0g‘𝐺) ∈ V | |
| 50 | 49 | ensn1 9062 | . 2 ⊢ {(0g‘𝐺)} ≈ 1o | 
| 51 | 48, 50 | eqbrtri 5163 | 1 ⊢ 𝐵 ≈ 1o | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 ∀wral 3060 ∃!wreu 3377 ∃*wrmo 3378 Vcvv 3479 ⊆ wss 3950 ∅c0 4332 {csn 4625 class class class wbr 5142 ↦ cmpt 5224 I cid 5576 × cxp 5682 ↾ cres 5686 ∘ ccom 5688 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 1oc1o 8500 ≈ cen 8983 Basecbs 17248 0gc0g 17485 Grpcgrp 18952 GrpHom cghm 19231 ~FG cefg 19725 freeGrpcfrgp 19726 varFGrpcvrgp 19727 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-word 14554 df-lsw 14602 df-concat 14610 df-s1 14635 df-substr 14680 df-pfx 14710 df-splice 14789 df-reverse 14798 df-s2 14888 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 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 17487 df-gsum 17488 df-imas 17554 df-qus 17555 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-frmd 18863 df-vrmd 18864 df-grp 18955 df-minusg 18956 df-ghm 19232 df-efg 19728 df-frgp 19729 df-vrgp 19730 | 
| This theorem is referenced by: frgpcyg 21593 | 
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