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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 17767 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd) |
6 | eqid 2795 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | dsmm0cl.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
8 | 6, 7 | mndidcl 17752 | . . 3 ⊢ (𝑃 ∈ Mnd → 0 ∈ (Base‘𝑃)) |
9 | 5, 8 | syl 17 | . 2 ⊢ (𝜑 → 0 ∈ (Base‘𝑃)) |
10 | 1, 2, 3, 4 | prds0g 17768 | . . . . . . . . . 10 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑃)) |
11 | 10, 7 | syl6eqr 2849 | . . . . . . . . 9 ⊢ (𝜑 → (0g ∘ 𝑅) = 0 ) |
12 | 11 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (0g ∘ 𝑅) = 0 ) |
13 | 12 | fveq1d 6545 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = ( 0 ‘𝑎)) |
14 | 4 | ffnd 6388 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
15 | fvco2 6630 | . . . . . . . 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 2833 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ( 0 ‘𝑎) = (0g‘(𝑅‘𝑎))) |
18 | nne 2988 | . . . . . 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 3149 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐼 ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) |
21 | rabeq0 4262 | . . . 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 8597 | . . 3 ⊢ ∅ ∈ Fin | |
24 | 22, 23 | syl6eqel 2891 | . 2 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin) |
25 | eqid 2795 | . . 3 ⊢ (𝑆 ⊕m 𝑅) = (𝑆 ⊕m 𝑅) | |
26 | dsmmcl.h | . . 3 ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) | |
27 | 1, 25, 6, 26, 2, 14 | dsmmelbas 20570 | . 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 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 {crab 3109 ∅c0 4215 ∘ ccom 5452 Fn wfn 6225 ⟶wf 6226 ‘cfv 6230 (class class class)co 7021 Fincfn 8362 Basecbs 16317 0gc0g 16547 Xscprds 16553 Mndcmnd 17738 ⊕m cdsmm 20562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-sup 8757 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-fz 12748 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-hom 16423 df-cco 16424 df-0g 16549 df-prds 16555 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-dsmm 20563 |
This theorem is referenced by: dsmmsubg 20574 |
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