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Mirrors > Home > MPE Home > Th. List > 0subgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of 0subg 19067 as of 31-Jan-2025. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0subg.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
0subgOLD | ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2730 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 0subg.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | grpidcl 18886 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
4 | 3 | snssd 4811 | . 2 ⊢ (𝐺 ∈ Grp → { 0 } ⊆ (Base‘𝐺)) |
5 | 2 | fvexi 6904 | . . . 4 ⊢ 0 ∈ V |
6 | 5 | snnz 4779 | . . 3 ⊢ { 0 } ≠ ∅ |
7 | 6 | a1i 11 | . 2 ⊢ (𝐺 ∈ Grp → { 0 } ≠ ∅) |
8 | eqid 2730 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
9 | 1, 8, 2 | grplid 18888 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
10 | 3, 9 | mpdan 683 | . . . 4 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) = 0 ) |
11 | ovex 7444 | . . . . 5 ⊢ ( 0 (+g‘𝐺) 0 ) ∈ V | |
12 | 11 | elsn 4642 | . . . 4 ⊢ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) = 0 ) |
13 | 10, 12 | sylibr 233 | . . 3 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
14 | eqid 2730 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
15 | 2, 14 | grpinvid 18920 | . . . 4 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
16 | fvex 6903 | . . . . 5 ⊢ ((invg‘𝐺)‘ 0 ) ∈ V | |
17 | 16 | elsn 4642 | . . . 4 ⊢ (((invg‘𝐺)‘ 0 ) ∈ { 0 } ↔ ((invg‘𝐺)‘ 0 ) = 0 ) |
18 | 15, 17 | sylibr 233 | . . 3 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) ∈ { 0 }) |
19 | oveq1 7418 | . . . . . . . 8 ⊢ (𝑎 = 0 → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺)𝑏)) | |
20 | 19 | eleq1d 2816 | . . . . . . 7 ⊢ (𝑎 = 0 → ((𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺)𝑏) ∈ { 0 })) |
21 | 20 | ralbidv 3175 | . . . . . 6 ⊢ (𝑎 = 0 → (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ∀𝑏 ∈ { 0 } ( 0 (+g‘𝐺)𝑏) ∈ { 0 })) |
22 | oveq2 7419 | . . . . . . . 8 ⊢ (𝑏 = 0 → ( 0 (+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 )) | |
23 | 22 | eleq1d 2816 | . . . . . . 7 ⊢ (𝑏 = 0 → (( 0 (+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
24 | 5, 23 | ralsn 4684 | . . . . . 6 ⊢ (∀𝑏 ∈ { 0 } ( 0 (+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
25 | 21, 24 | bitrdi 286 | . . . . 5 ⊢ (𝑎 = 0 → (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
26 | fveq2 6890 | . . . . . 6 ⊢ (𝑎 = 0 → ((invg‘𝐺)‘𝑎) = ((invg‘𝐺)‘ 0 )) | |
27 | 26 | eleq1d 2816 | . . . . 5 ⊢ (𝑎 = 0 → (((invg‘𝐺)‘𝑎) ∈ { 0 } ↔ ((invg‘𝐺)‘ 0 ) ∈ { 0 })) |
28 | 25, 27 | anbi12d 629 | . . . 4 ⊢ (𝑎 = 0 → ((∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 }) ↔ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ∧ ((invg‘𝐺)‘ 0 ) ∈ { 0 }))) |
29 | 5, 28 | ralsn 4684 | . . 3 ⊢ (∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 }) ↔ (( 0 (+g‘𝐺) 0 ) ∈ { 0 } ∧ ((invg‘𝐺)‘ 0 ) ∈ { 0 })) |
30 | 13, 18, 29 | sylanbrc 581 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 })) |
31 | 1, 8, 14 | issubg2 19057 | . 2 ⊢ (𝐺 ∈ Grp → ({ 0 } ∈ (SubGrp‘𝐺) ↔ ({ 0 } ⊆ (Base‘𝐺) ∧ { 0 } ≠ ∅ ∧ ∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 })))) |
32 | 4, 7, 30, 31 | mpbir3and 1340 | 1 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ⊆ wss 3947 ∅c0 4321 {csn 4627 ‘cfv 6542 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 0gc0g 17389 Grpcgrp 18855 invgcminusg 18856 SubGrpcsubg 19036 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-subg 19039 |
This theorem is referenced by: (None) |
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