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Mirrors > Home > MPE Home > Th. List > dsmm0cl | Structured version Visualization version GIF version |
Description: The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
Ref | Expression |
---|---|
dsmmcl.p | ⊢ 𝑃 = (𝑆Xs𝑅) |
dsmmcl.h | ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) |
dsmmcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
dsmmcl.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
dsmmcl.r | ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
dsmm0cl.z | ⊢ 0 = (0g‘𝑃) |
Ref | Expression |
---|---|
dsmm0cl | ⊢ (𝜑 → 0 ∈ 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dsmmcl.p | . . . 4 ⊢ 𝑃 = (𝑆Xs𝑅) | |
2 | dsmmcl.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
3 | dsmmcl.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | dsmmcl.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) | |
5 | 1, 2, 3, 4 | prdsmndd 17323 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd) |
6 | eqid 2622 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | dsmm0cl.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
8 | 6, 7 | mndidcl 17308 | . . 3 ⊢ (𝑃 ∈ Mnd → 0 ∈ (Base‘𝑃)) |
9 | 5, 8 | syl 17 | . 2 ⊢ (𝜑 → 0 ∈ (Base‘𝑃)) |
10 | 1, 2, 3, 4 | prds0g 17324 | . . . . . . . . . 10 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑃)) |
11 | 10, 7 | syl6eqr 2674 | . . . . . . . . 9 ⊢ (𝜑 → (0g ∘ 𝑅) = 0 ) |
12 | 11 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (0g ∘ 𝑅) = 0 ) |
13 | 12 | fveq1d 6193 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = ( 0 ‘𝑎)) |
14 | ffn 6045 | . . . . . . . . 9 ⊢ (𝑅:𝐼⟶Mnd → 𝑅 Fn 𝐼) | |
15 | 4, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
16 | fvco2 6273 | . . . . . . . 8 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = (0g‘(𝑅‘𝑎))) | |
17 | 15, 16 | sylan 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = (0g‘(𝑅‘𝑎))) |
18 | 13, 17 | eqtr3d 2658 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ( 0 ‘𝑎) = (0g‘(𝑅‘𝑎))) |
19 | nne 2798 | . . . . . 6 ⊢ (¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎)) ↔ ( 0 ‘𝑎) = (0g‘(𝑅‘𝑎))) | |
20 | 18, 19 | sylibr 224 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) |
21 | 20 | ralrimiva 2966 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐼 ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) |
22 | rabeq0 3957 | . . . 4 ⊢ ({𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = ∅ ↔ ∀𝑎 ∈ 𝐼 ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) | |
23 | 21, 22 | sylibr 224 | . . 3 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = ∅) |
24 | 0fin 8188 | . . 3 ⊢ ∅ ∈ Fin | |
25 | 23, 24 | syl6eqel 2709 | . 2 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin) |
26 | eqid 2622 | . . 3 ⊢ (𝑆 ⊕m 𝑅) = (𝑆 ⊕m 𝑅) | |
27 | dsmmcl.h | . . 3 ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) | |
28 | 1, 26, 6, 27, 2, 15 | dsmmelbas 20083 | . 2 ⊢ (𝜑 → ( 0 ∈ 𝐻 ↔ ( 0 ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) |
29 | 9, 25, 28 | mpbir2and 957 | 1 ⊢ (𝜑 → 0 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ∅c0 3915 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 Basecbs 15857 0gc0g 16100 Xscprds 16106 Mndcmnd 17294 ⊕m cdsmm 20075 |
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-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 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-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-dsmm 20076 |
This theorem is referenced by: dsmmsubg 20087 |
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