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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmsuppss | Structured version Visualization version GIF version |
Description: The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.) |
Ref | Expression |
---|---|
rmsuppss.r | ⊢ 𝑅 = (Base‘𝑀) |
Ref | Expression |
---|---|
rmsuppss | ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7143 | . . . . . . 7 ⊢ ((𝐴‘𝑤) = (0g‘𝑀) → (𝐶(.r‘𝑀)(𝐴‘𝑤)) = (𝐶(.r‘𝑀)(0g‘𝑀))) | |
2 | simpll1 1209 | . . . . . . . 8 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → 𝑀 ∈ Ring) | |
3 | simpll3 1211 | . . . . . . . 8 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → 𝐶 ∈ 𝑅) | |
4 | rmsuppss.r | . . . . . . . . 9 ⊢ 𝑅 = (Base‘𝑀) | |
5 | eqid 2798 | . . . . . . . . 9 ⊢ (.r‘𝑀) = (.r‘𝑀) | |
6 | eqid 2798 | . . . . . . . . 9 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
7 | 4, 5, 6 | ringrz 19334 | . . . . . . . 8 ⊢ ((𝑀 ∈ Ring ∧ 𝐶 ∈ 𝑅) → (𝐶(.r‘𝑀)(0g‘𝑀)) = (0g‘𝑀)) |
8 | 2, 3, 7 | syl2anc 587 | . . . . . . 7 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → (𝐶(.r‘𝑀)(0g‘𝑀)) = (0g‘𝑀)) |
9 | 1, 8 | sylan9eqr 2855 | . . . . . 6 ⊢ (((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) ∧ (𝐴‘𝑤) = (0g‘𝑀)) → (𝐶(.r‘𝑀)(𝐴‘𝑤)) = (0g‘𝑀)) |
10 | 9 | ex 416 | . . . . 5 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → ((𝐴‘𝑤) = (0g‘𝑀) → (𝐶(.r‘𝑀)(𝐴‘𝑤)) = (0g‘𝑀))) |
11 | 10 | necon3d 3008 | . . . 4 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → ((𝐶(.r‘𝑀)(𝐴‘𝑤)) ≠ (0g‘𝑀) → (𝐴‘𝑤) ≠ (0g‘𝑀))) |
12 | 11 | ss2rabdv 4003 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → {𝑤 ∈ 𝑉 ∣ (𝐶(.r‘𝑀)(𝐴‘𝑤)) ≠ (0g‘𝑀)} ⊆ {𝑤 ∈ 𝑉 ∣ (𝐴‘𝑤) ≠ (0g‘𝑀)}) |
13 | elmapi 8411 | . . . . . 6 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅) | |
14 | 13 | fdmd 6497 | . . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → dom 𝐴 = 𝑉) |
15 | 14 | adantl 485 | . . . 4 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → dom 𝐴 = 𝑉) |
16 | rabeq 3431 | . . . 4 ⊢ (dom 𝐴 = 𝑉 → {𝑤 ∈ dom 𝐴 ∣ (𝐴‘𝑤) ≠ (0g‘𝑀)} = {𝑤 ∈ 𝑉 ∣ (𝐴‘𝑤) ≠ (0g‘𝑀)}) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → {𝑤 ∈ dom 𝐴 ∣ (𝐴‘𝑤) ≠ (0g‘𝑀)} = {𝑤 ∈ 𝑉 ∣ (𝐴‘𝑤) ≠ (0g‘𝑀)}) |
18 | 12, 17 | sseqtrrd 3956 | . 2 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → {𝑤 ∈ 𝑉 ∣ (𝐶(.r‘𝑀)(𝐴‘𝑤)) ≠ (0g‘𝑀)} ⊆ {𝑤 ∈ dom 𝐴 ∣ (𝐴‘𝑤) ≠ (0g‘𝑀)}) |
19 | fveq2 6645 | . . . . 5 ⊢ (𝑣 = 𝑤 → (𝐴‘𝑣) = (𝐴‘𝑤)) | |
20 | 19 | oveq2d 7151 | . . . 4 ⊢ (𝑣 = 𝑤 → (𝐶(.r‘𝑀)(𝐴‘𝑣)) = (𝐶(.r‘𝑀)(𝐴‘𝑤))) |
21 | 20 | cbvmptv 5133 | . . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) = (𝑤 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑤))) |
22 | simpl2 1189 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝑉 ∈ 𝑋) | |
23 | fvexd 6660 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (0g‘𝑀) ∈ V) | |
24 | ovexd 7170 | . . 3 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → (𝐶(.r‘𝑀)(𝐴‘𝑤)) ∈ V) | |
25 | 21, 22, 23, 24 | mptsuppd 7836 | . 2 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) = {𝑤 ∈ 𝑉 ∣ (𝐶(.r‘𝑀)(𝐴‘𝑤)) ≠ (0g‘𝑀)}) |
26 | elmapfun 8413 | . . . 4 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → Fun 𝐴) | |
27 | 26 | adantl 485 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → Fun 𝐴) |
28 | simpr 488 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝐴 ∈ (𝑅 ↑m 𝑉)) | |
29 | suppval1 7819 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (0g‘𝑀) ∈ V) → (𝐴 supp (0g‘𝑀)) = {𝑤 ∈ dom 𝐴 ∣ (𝐴‘𝑤) ≠ (0g‘𝑀)}) | |
30 | 27, 28, 23, 29 | syl3anc 1368 | . 2 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (𝐴 supp (0g‘𝑀)) = {𝑤 ∈ dom 𝐴 ∣ (𝐴‘𝑤) ≠ (0g‘𝑀)}) |
31 | 18, 25, 30 | 3sstr4d 3962 | 1 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 Vcvv 3441 ⊆ wss 3881 ↦ cmpt 5110 dom cdm 5519 Fun wfun 6318 ‘cfv 6324 (class class class)co 7135 supp csupp 7813 ↑m cmap 8389 Basecbs 16475 .rcmulr 16558 0gc0g 16705 Ringcrg 19290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-mgp 19233 df-ring 19292 |
This theorem is referenced by: rmsuppfi 44775 |
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