![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 18645 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd) |
6 | eqid 2733 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | dsmm0cl.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
8 | 6, 7 | mndidcl 18627 | . . 3 ⊢ (𝑃 ∈ Mnd → 0 ∈ (Base‘𝑃)) |
9 | 5, 8 | syl 17 | . 2 ⊢ (𝜑 → 0 ∈ (Base‘𝑃)) |
10 | 1, 2, 3, 4 | prds0g 18646 | . . . . . . . . . 10 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑃)) |
11 | 10, 7 | eqtr4di 2791 | . . . . . . . . 9 ⊢ (𝜑 → (0g ∘ 𝑅) = 0 ) |
12 | 11 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → (0g ∘ 𝑅) = 0 ) |
13 | 12 | fveq1d 6883 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = ( 0 ‘𝑎)) |
14 | 4 | ffnd 6708 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
15 | fvco2 6977 | . . . . . . . 8 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = (0g‘(𝑅‘𝑎))) | |
16 | 14, 15 | sylan 581 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑎) = (0g‘(𝑅‘𝑎))) |
17 | 13, 16 | eqtr3d 2775 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ( 0 ‘𝑎) = (0g‘(𝑅‘𝑎))) |
18 | nne 2945 | . . . . . 6 ⊢ (¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎)) ↔ ( 0 ‘𝑎) = (0g‘(𝑅‘𝑎))) | |
19 | 17, 18 | sylibr 233 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) |
20 | 19 | ralrimiva 3147 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐼 ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) |
21 | rabeq0 4382 | . . . 4 ⊢ ({𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = ∅ ↔ ∀𝑎 ∈ 𝐼 ¬ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))) | |
22 | 20, 21 | sylibr 233 | . . 3 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} = ∅) |
23 | 0fin 9159 | . . 3 ⊢ ∅ ∈ Fin | |
24 | 22, 23 | eqeltrdi 2842 | . 2 ⊢ (𝜑 → {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin) |
25 | eqid 2733 | . . 3 ⊢ (𝑆 ⊕m 𝑅) = (𝑆 ⊕m 𝑅) | |
26 | dsmmcl.h | . . 3 ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) | |
27 | 1, 25, 6, 26, 2, 14 | dsmmelbas 21267 | . 2 ⊢ (𝜑 → ( 0 ∈ 𝐻 ↔ ( 0 ∈ (Base‘𝑃) ∧ {𝑎 ∈ 𝐼 ∣ ( 0 ‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) |
28 | 9, 24, 27 | mpbir2and 712 | 1 ⊢ (𝜑 → 0 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 {crab 3433 ∅c0 4320 ∘ ccom 5676 Fn wfn 6530 ⟶wf 6531 ‘cfv 6535 (class class class)co 7396 Fincfn 8927 Basecbs 17131 0gc0g 17372 Xscprds 17378 Mndcmnd 18612 ⊕m cdsmm 21259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9424 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-fz 13472 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-hom 17208 df-cco 17209 df-0g 17374 df-prds 17380 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-dsmm 21260 |
This theorem is referenced by: dsmmsubg 21271 |
Copyright terms: Public domain | W3C validator |