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 7158 | . . . . . . 7 ⊢ ((𝐴‘𝑤) = (0g‘𝑀) → (𝐶(.r‘𝑀)(𝐴‘𝑤)) = (𝐶(.r‘𝑀)(0g‘𝑀))) | |
2 | simpll1 1208 | . . . . . . . 8 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → 𝑀 ∈ Ring) | |
3 | simpll3 1210 | . . . . . . . 8 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → 𝐶 ∈ 𝑅) | |
4 | rmsuppss.r | . . . . . . . . 9 ⊢ 𝑅 = (Base‘𝑀) | |
5 | eqid 2821 | . . . . . . . . 9 ⊢ (.r‘𝑀) = (.r‘𝑀) | |
6 | eqid 2821 | . . . . . . . . 9 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
7 | 4, 5, 6 | ringrz 19332 | . . . . . . . 8 ⊢ ((𝑀 ∈ Ring ∧ 𝐶 ∈ 𝑅) → (𝐶(.r‘𝑀)(0g‘𝑀)) = (0g‘𝑀)) |
8 | 2, 3, 7 | syl2anc 586 | . . . . . . 7 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → (𝐶(.r‘𝑀)(0g‘𝑀)) = (0g‘𝑀)) |
9 | 1, 8 | sylan9eqr 2878 | . . . . . 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 415 | . . . . 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 3037 | . . . 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 4052 | . . 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 8422 | . . . . . 6 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅) | |
14 | 13 | fdmd 6518 | . . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → dom 𝐴 = 𝑉) |
15 | 14 | adantl 484 | . . . 4 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → dom 𝐴 = 𝑉) |
16 | rabeq 3484 | . . . 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 4008 | . 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 6665 | . . . . 5 ⊢ (𝑣 = 𝑤 → (𝐴‘𝑣) = (𝐴‘𝑤)) | |
20 | 19 | oveq2d 7166 | . . . 4 ⊢ (𝑣 = 𝑤 → (𝐶(.r‘𝑀)(𝐴‘𝑣)) = (𝐶(.r‘𝑀)(𝐴‘𝑤))) |
21 | 20 | cbvmptv 5162 | . . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) = (𝑤 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑤))) |
22 | simpl2 1188 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝑉 ∈ 𝑋) | |
23 | fvexd 6680 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (0g‘𝑀) ∈ V) | |
24 | ovexd 7185 | . . 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 7847 | . 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 8424 | . . . 4 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → Fun 𝐴) | |
27 | 26 | adantl 484 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → Fun 𝐴) |
28 | simpr 487 | . . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝐴 ∈ (𝑅 ↑m 𝑉)) | |
29 | suppval1 7830 | . . 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 1367 | . 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 4014 | 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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {crab 3142 Vcvv 3495 ⊆ wss 3936 ↦ cmpt 5139 dom cdm 5550 Fun wfun 6344 ‘cfv 6350 (class class class)co 7150 supp csupp 7824 ↑m cmap 8400 Basecbs 16477 .rcmulr 16560 0gc0g 16707 Ringcrg 19291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-mgp 19234 df-ring 19293 |
This theorem is referenced by: rmsuppfi 44414 |
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