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Mirrors > Home > MPE Home > Th. List > obs2ss | Structured version Visualization version GIF version |
Description: A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obs2ss | ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1063 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵) | |
2 | eqid 2622 | . . . . . . 7 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
3 | 2 | obsne0 20069 | . . . . . 6 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑥 ∈ 𝐵) → 𝑥 ≠ (0g‘𝑊)) |
4 | 3 | 3ad2antl1 1223 | . . . . 5 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ≠ (0g‘𝑊)) |
5 | eqid 2622 | . . . . . . . . . 10 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
6 | 5 | obselocv 20072 | . . . . . . . . 9 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ ((ocv‘𝑊)‘𝐶) ↔ ¬ 𝑥 ∈ 𝐶)) |
7 | 6 | 3expa 1265 | . . . . . . . 8 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ ((ocv‘𝑊)‘𝐶) ↔ ¬ 𝑥 ∈ 𝐶)) |
8 | 7 | 3adantl2 1218 | . . . . . . 7 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ ((ocv‘𝑊)‘𝐶) ↔ ¬ 𝑥 ∈ 𝐶)) |
9 | simpl2 1065 | . . . . . . . . . 10 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ (OBasis‘𝑊)) | |
10 | 2, 5 | obsocv 20070 | . . . . . . . . . 10 ⊢ (𝐶 ∈ (OBasis‘𝑊) → ((ocv‘𝑊)‘𝐶) = {(0g‘𝑊)}) |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((ocv‘𝑊)‘𝐶) = {(0g‘𝑊)}) |
12 | 11 | eleq2d 2687 | . . . . . . . 8 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ ((ocv‘𝑊)‘𝐶) ↔ 𝑥 ∈ {(0g‘𝑊)})) |
13 | elsni 4194 | . . . . . . . 8 ⊢ (𝑥 ∈ {(0g‘𝑊)} → 𝑥 = (0g‘𝑊)) | |
14 | 12, 13 | syl6bi 243 | . . . . . . 7 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ ((ocv‘𝑊)‘𝐶) → 𝑥 = (0g‘𝑊))) |
15 | 8, 14 | sylbird 250 | . . . . . 6 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → (¬ 𝑥 ∈ 𝐶 → 𝑥 = (0g‘𝑊))) |
16 | 15 | necon1ad 2811 | . . . . 5 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ≠ (0g‘𝑊) → 𝑥 ∈ 𝐶)) |
17 | 4, 16 | mpd 15 | . . . 4 ⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) |
18 | 17 | ex 450 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) |
19 | 18 | ssrdv 3609 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) → 𝐵 ⊆ 𝐶) |
20 | 1, 19 | eqssd 3620 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 {csn 4177 ‘cfv 5888 0gc0g 16100 ocvcocv 20004 OBasiscobs 20046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-ghm 17658 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-rnghom 18715 df-drng 18749 df-staf 18845 df-srng 18846 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lmhm 19022 df-lvec 19103 df-sra 19172 df-rgmod 19173 df-phl 19971 df-ocv 20007 df-obs 20049 |
This theorem is referenced by: (None) |
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