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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 17946 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd) |
6 | eqid 2823 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | dsmm0cl.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
8 | 6, 7 | mndidcl 17928 | . . 3 ⊢ (𝑃 ∈ Mnd → 0 ∈ (Base‘𝑃)) |
9 | 5, 8 | syl 17 | . 2 ⊢ (𝜑 → 0 ∈ (Base‘𝑃)) |
10 | 1, 2, 3, 4 | prds0g 17947 | . . . . . . . . . 10 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑃)) |
11 | 10, 7 | syl6eqr 2876 | . . . . . . . . 9 ⊢ (𝜑 → (0g ∘ 𝑅) = 0 ) |
12 | 11 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (0g ∘ 𝑅) = 0 ) |
13 | 12 | fveq1d 6674 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = ( 0 ‘𝑎)) |
14 | 4 | ffnd 6517 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
15 | fvco2 6760 | . . . . . . . 8 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = (0g‘(𝑅‘𝑎))) | |
16 | 14, 15 | sylan 582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = (0g‘(𝑅‘𝑎))) |
17 | 13, 16 | eqtr3d 2860 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ( 0 ‘𝑎) = (0g‘(𝑅‘𝑎))) |
18 | nne 3022 | . . . . . 6 ⊢ (¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎)) ↔ ( 0 ‘𝑎) = (0g‘(𝑅‘𝑎))) | |
19 | 17, 18 | sylibr 236 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) |
20 | 19 | ralrimiva 3184 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐼 ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) |
21 | rabeq0 4340 | . . . 4 ⊢ ({𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = ∅ ↔ ∀𝑎 ∈ 𝐼 ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) | |
22 | 20, 21 | sylibr 236 | . . 3 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = ∅) |
23 | 0fin 8748 | . . 3 ⊢ ∅ ∈ Fin | |
24 | 22, 23 | eqeltrdi 2923 | . 2 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin) |
25 | eqid 2823 | . . 3 ⊢ (𝑆 ⊕m 𝑅) = (𝑆 ⊕m 𝑅) | |
26 | dsmmcl.h | . . 3 ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) | |
27 | 1, 25, 6, 26, 2, 14 | dsmmelbas 20885 | . 2 ⊢ (𝜑 → ( 0 ∈ 𝐻 ↔ ( 0 ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) |
28 | 9, 24, 27 | mpbir2and 711 | 1 ⊢ (𝜑 → 0 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 {crab 3144 ∅c0 4293 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 Basecbs 16485 0gc0g 16715 Xscprds 16721 Mndcmnd 17913 ⊕m cdsmm 20877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-prds 16723 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-dsmm 20878 |
This theorem is referenced by: dsmmsubg 20889 |
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