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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 17932 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd) |
6 | eqid 2818 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | dsmm0cl.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
8 | 6, 7 | mndidcl 17914 | . . 3 ⊢ (𝑃 ∈ Mnd → 0 ∈ (Base‘𝑃)) |
9 | 5, 8 | syl 17 | . 2 ⊢ (𝜑 → 0 ∈ (Base‘𝑃)) |
10 | 1, 2, 3, 4 | prds0g 17933 | . . . . . . . . . 10 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑃)) |
11 | 10, 7 | syl6eqr 2871 | . . . . . . . . 9 ⊢ (𝜑 → (0g ∘ 𝑅) = 0 ) |
12 | 11 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (0g ∘ 𝑅) = 0 ) |
13 | 12 | fveq1d 6665 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = ( 0 ‘𝑎)) |
14 | 4 | ffnd 6508 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
15 | fvco2 6751 | . . . . . . . 8 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = (0g‘(𝑅‘𝑎))) | |
16 | 14, 15 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = (0g‘(𝑅‘𝑎))) |
17 | 13, 16 | eqtr3d 2855 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ( 0 ‘𝑎) = (0g‘(𝑅‘𝑎))) |
18 | nne 3017 | . . . . . 6 ⊢ (¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎)) ↔ ( 0 ‘𝑎) = (0g‘(𝑅‘𝑎))) | |
19 | 17, 18 | sylibr 235 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) |
20 | 19 | ralrimiva 3179 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐼 ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) |
21 | rabeq0 4335 | . . . 4 ⊢ ({𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = ∅ ↔ ∀𝑎 ∈ 𝐼 ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) | |
22 | 20, 21 | sylibr 235 | . . 3 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = ∅) |
23 | 0fin 8734 | . . 3 ⊢ ∅ ∈ Fin | |
24 | 22, 23 | syl6eqel 2918 | . 2 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin) |
25 | eqid 2818 | . . 3 ⊢ (𝑆 ⊕m 𝑅) = (𝑆 ⊕m 𝑅) | |
26 | dsmmcl.h | . . 3 ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) | |
27 | 1, 25, 6, 26, 2, 14 | dsmmelbas 20811 | . 2 ⊢ (𝜑 → ( 0 ∈ 𝐻 ↔ ( 0 ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) |
28 | 9, 24, 27 | mpbir2and 709 | 1 ⊢ (𝜑 → 0 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 {crab 3139 ∅c0 4288 ∘ ccom 5552 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 Basecbs 16471 0gc0g 16701 Xscprds 16707 Mndcmnd 17899 ⊕m cdsmm 20803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-dsmm 20804 |
This theorem is referenced by: dsmmsubg 20815 |
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