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Mirrors > Home > MPE Home > Th. List > 0subgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of 0subg 19024 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 2733 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 0subg.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | grpidcl 18845 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺)) |
4 | 3 | snssd 4810 | . 2 ⊢ (𝐺 ∈ Grp → { 0 } ⊆ (Base‘𝐺)) |
5 | 2 | fvexi 6901 | . . . 4 ⊢ 0 ∈ V |
6 | 5 | snnz 4778 | . . 3 ⊢ { 0 } ≠ ∅ |
7 | 6 | a1i 11 | . 2 ⊢ (𝐺 ∈ Grp → { 0 } ≠ ∅) |
8 | eqid 2733 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
9 | 1, 8, 2 | grplid 18847 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
10 | 3, 9 | mpdan 686 | . . . 4 ⊢ (𝐺 ∈ Grp → ( 0 (+g‘𝐺) 0 ) = 0 ) |
11 | ovex 7436 | . . . . 5 ⊢ ( 0 (+g‘𝐺) 0 ) ∈ V | |
12 | 11 | elsn 4641 | . . . 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 2733 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
15 | 2, 14 | grpinvid 18879 | . . . 4 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
16 | fvex 6900 | . . . . 5 ⊢ ((invg‘𝐺)‘ 0 ) ∈ V | |
17 | 16 | elsn 4641 | . . . 4 ⊢ (((invg‘𝐺)‘ 0 ) ∈ { 0 } ↔ ((invg‘𝐺)‘ 0 ) = 0 ) |
18 | 15, 17 | sylibr 233 | . . 3 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) ∈ { 0 }) |
19 | oveq1 7410 | . . . . . . . 8 ⊢ (𝑎 = 0 → (𝑎(+g‘𝐺)𝑏) = ( 0 (+g‘𝐺)𝑏)) | |
20 | 19 | eleq1d 2819 | . . . . . . 7 ⊢ (𝑎 = 0 → ((𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺)𝑏) ∈ { 0 })) |
21 | 20 | ralbidv 3178 | . . . . . 6 ⊢ (𝑎 = 0 → (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ∀𝑏 ∈ { 0 } ( 0 (+g‘𝐺)𝑏) ∈ { 0 })) |
22 | oveq2 7411 | . . . . . . . 8 ⊢ (𝑏 = 0 → ( 0 (+g‘𝐺)𝑏) = ( 0 (+g‘𝐺) 0 )) | |
23 | 22 | eleq1d 2819 | . . . . . . 7 ⊢ (𝑏 = 0 → (( 0 (+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
24 | 5, 23 | ralsn 4683 | . . . . . 6 ⊢ (∀𝑏 ∈ { 0 } ( 0 (+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 }) |
25 | 21, 24 | bitrdi 287 | . . . . 5 ⊢ (𝑎 = 0 → (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ↔ ( 0 (+g‘𝐺) 0 ) ∈ { 0 })) |
26 | fveq2 6887 | . . . . . 6 ⊢ (𝑎 = 0 → ((invg‘𝐺)‘𝑎) = ((invg‘𝐺)‘ 0 )) | |
27 | 26 | eleq1d 2819 | . . . . 5 ⊢ (𝑎 = 0 → (((invg‘𝐺)‘𝑎) ∈ { 0 } ↔ ((invg‘𝐺)‘ 0 ) ∈ { 0 })) |
28 | 25, 27 | anbi12d 632 | . . . 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 4683 | . . 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 584 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ { 0 } (∀𝑏 ∈ { 0 } (𝑎(+g‘𝐺)𝑏) ∈ { 0 } ∧ ((invg‘𝐺)‘𝑎) ∈ { 0 })) |
31 | 1, 8, 14 | issubg2 19014 | . 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 1343 | 1 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ⊆ wss 3946 ∅c0 4320 {csn 4626 ‘cfv 6539 (class class class)co 7403 Basecbs 17139 +gcplusg 17192 0gc0g 17380 Grpcgrp 18814 invgcminusg 18815 SubGrpcsubg 18993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-0g 17382 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-grp 18817 df-minusg 18818 df-subg 18996 |
This theorem is referenced by: (None) |
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