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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 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 ⊢ (varFGrp‘∅) = (varFGrp‘∅) | |
9 | 7, 8, 2, 6 | vrgpf 19801 | . . . . . . . . . . . . . . 15 ⊢ (∅ ∈ V → (varFGrp‘∅):∅⟶𝐵) |
10 | ffn 6737 | . . . . . . . . . . . . . . 15 ⊢ ((varFGrp‘∅):∅⟶𝐵 → (varFGrp‘∅) Fn ∅) | |
11 | 1, 9, 10 | mp2b 10 | . . . . . . . . . . . . . 14 ⊢ (varFGrp‘∅) Fn ∅ |
12 | fn0 6700 | . . . . . . . . . . . . . 14 ⊢ ((varFGrp‘∅) Fn ∅ ↔ (varFGrp‘∅) = ∅) | |
13 | 11, 12 | mpbi 230 | . . . . . . . . . . . . 13 ⊢ (varFGrp‘∅) = ∅ |
14 | 13 | eqcomi 2744 | . . . . . . . . . . . 12 ⊢ ∅ = (varFGrp‘∅) |
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 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
24 | 23, 6 | 0ghm 19261 | . . . . . . . . . . 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 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 × {(0g‘𝐺)}) → ((𝑓 ∘ ∅) = ∅ ↔ ⊤)) |
31 | 29, 30 | rmoi 3900 | . . . . . . . . 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 1460 | . . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝐵 × {(0g‘𝐺)}) |
33 | mptresid 6071 | . . . . . . . 8 ⊢ ( I ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝑥) | |
34 | fconstmpt 5751 | . . . . . . . 8 ⊢ (𝐵 × {(0g‘𝐺)}) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) | |
35 | 32, 33, 34 | 3eqtr3i 2771 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑥) = (𝑥 ∈ 𝐵 ↦ (0g‘𝐺)) |
36 | mpteqb 7035 | . . . . . . . 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 3065 | . . . . . . 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 3248 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0g‘𝐺)) |
41 | velsn 4647 | . . . . 5 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
42 | 40, 41 | sylibr 234 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {(0g‘𝐺)}) |
43 | 42 | ssriv 3999 | . . 3 ⊢ 𝐵 ⊆ {(0g‘𝐺)} |
44 | 6, 23 | grpidcl 18996 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) |
45 | 4, 44 | ax-mp 5 | . . . 4 ⊢ (0g‘𝐺) ∈ 𝐵 |
46 | snssi 4813 | . . . 4 ⊢ ((0g‘𝐺) ∈ 𝐵 → {(0g‘𝐺)} ⊆ 𝐵) | |
47 | 45, 46 | ax-mp 5 | . . 3 ⊢ {(0g‘𝐺)} ⊆ 𝐵 |
48 | 43, 47 | eqssi 4012 | . 2 ⊢ 𝐵 = {(0g‘𝐺)} |
49 | fvex 6920 | . . 3 ⊢ (0g‘𝐺) ∈ V | |
50 | 49 | ensn1 9060 | . 2 ⊢ {(0g‘𝐺)} ≈ 1o |
51 | 48, 50 | eqbrtri 5169 | 1 ⊢ 𝐵 ≈ 1o |
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 1oc1o 8498 ≈ cen 8981 Basecbs 17245 0gc0g 17486 Grpcgrp 18964 GrpHom cghm 19243 ~FG cefg 19739 freeGrpcfrgp 19740 varFGrpcvrgp 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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