Theorem rmsupp0 47044

Description: The support of a mapping of a multiplication of zero with a function into a ring is empty. (Contributed by AV, 10-Apr-2019.)

Hypothesis

Ref	Expression
rmsuppss.r	⊢ 𝑅 = (Base‘𝑀)

Assertion

Ref	Expression
rmsupp0	⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) = ∅)

Distinct variable groups:   𝑣,𝐴   𝑣,𝐶   𝑣,𝑀   𝑣,𝑅   𝑣,𝑋   𝑣,𝑉

Proof of Theorem rmsupp0

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6892	. . . . 5 ⊢ (𝑣 = 𝑤 → (𝐴‘𝑣) = (𝐴‘𝑤))
2	1	oveq2d 7425	. . . 4 ⊢ (𝑣 = 𝑤 → (𝐶(.r‘𝑀)(𝐴‘𝑣)) = (𝐶(.r‘𝑀)(𝐴‘𝑤)))
3	2	cbvmptv 5262	. . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) = (𝑤 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑤)))
4		simpl2 1193	. . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝑉 ∈ 𝑋)
5		fvexd 6907	. . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (0g‘𝑀) ∈ V)
6		ovexd 7444	. . 3 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → (𝐶(.r‘𝑀)(𝐴‘𝑤)) ∈ V)
7	3, 4, 5, 6	mptsuppd 8172	. 2 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) = {𝑤 ∈ 𝑉 ∣ (𝐶(.r‘𝑀)(𝐴‘𝑤)) ≠ (0g‘𝑀)})
8		simpll3 1215	. . . . . 6 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → 𝐶 = (0g‘𝑀))
9	8	oveq1d 7424	. . . . 5 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → (𝐶(.r‘𝑀)(𝐴‘𝑤)) = ((0g‘𝑀)(.r‘𝑀)(𝐴‘𝑤)))
10		simpll1 1213	. . . . . 6 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → 𝑀 ∈ Ring)
11		elmapi 8843	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
12		ffvelcdm 7084	. . . . . . . . . . 11 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑤 ∈ 𝑉) → (𝐴‘𝑤) ∈ 𝑅)
13		rmsuppss.r	. . . . . . . . . . 11 ⊢ 𝑅 = (Base‘𝑀)
14	12, 13	eleqtrdi 2844	. . . . . . . . . 10 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑤 ∈ 𝑉) → (𝐴‘𝑤) ∈ (Base‘𝑀))
15	14	ex 414	. . . . . . . . 9 ⊢ (𝐴:𝑉⟶𝑅 → (𝑤 ∈ 𝑉 → (𝐴‘𝑤) ∈ (Base‘𝑀)))
16	11, 15	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑤 ∈ 𝑉 → (𝐴‘𝑤) ∈ (Base‘𝑀)))
17	16	adantl 483	. . . . . . 7 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (𝑤 ∈ 𝑉 → (𝐴‘𝑤) ∈ (Base‘𝑀)))
18	17	imp 408	. . . . . 6 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → (𝐴‘𝑤) ∈ (Base‘𝑀))
19		eqid 2733	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
20		eqid 2733	. . . . . . 7 ⊢ (.r‘𝑀) = (.r‘𝑀)
21		eqid 2733	. . . . . . 7 ⊢ (0g‘𝑀) = (0g‘𝑀)
22	19, 20, 21	ringlz 20107	. . . . . 6 ⊢ ((𝑀 ∈ Ring ∧ (𝐴‘𝑤) ∈ (Base‘𝑀)) → ((0g‘𝑀)(.r‘𝑀)(𝐴‘𝑤)) = (0g‘𝑀))
23	10, 18, 22	syl2anc 585	. . . . 5 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → ((0g‘𝑀)(.r‘𝑀)(𝐴‘𝑤)) = (0g‘𝑀))
24	9, 23	eqtrd 2773	. . . 4 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → (𝐶(.r‘𝑀)(𝐴‘𝑤)) = (0g‘𝑀))
25	24	neeq1d 3001	. . 3 ⊢ ((((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) ∧ 𝑤 ∈ 𝑉) → ((𝐶(.r‘𝑀)(𝐴‘𝑤)) ≠ (0g‘𝑀) ↔ (0g‘𝑀) ≠ (0g‘𝑀)))
26	25	rabbidva 3440	. 2 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → {𝑤 ∈ 𝑉 ∣ (𝐶(.r‘𝑀)(𝐴‘𝑤)) ≠ (0g‘𝑀)} = {𝑤 ∈ 𝑉 ∣ (0g‘𝑀) ≠ (0g‘𝑀)})
27		neirr 2950	. . . . 5 ⊢ ¬ (0g‘𝑀) ≠ (0g‘𝑀)
28	27	a1i 11	. . . 4 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ¬ (0g‘𝑀) ≠ (0g‘𝑀))
29	28	ralrimivw 3151	. . 3 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ∀𝑤 ∈ 𝑉 ¬ (0g‘𝑀) ≠ (0g‘𝑀))
30		rabeq0 4385	. . 3 ⊢ ({𝑤 ∈ 𝑉 ∣ (0g‘𝑀) ≠ (0g‘𝑀)} = ∅ ↔ ∀𝑤 ∈ 𝑉 ¬ (0g‘𝑀) ≠ (0g‘𝑀))
31	29, 30	sylibr 233	. 2 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → {𝑤 ∈ 𝑉 ∣ (0g‘𝑀) ≠ (0g‘𝑀)} = ∅)
32	7, 26, 31	3eqtrd 2777	1 ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  {crab 3433  Vcvv 3475  ∅c0 4323   ↦ cmpt 5232  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409   supp csupp 8146   ↑_m cmap 8820  Basecbs 17144  ._rcmulr 17198  0_gc0g 17385  Ringcrg 20056

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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-mgp 19988  df-ring 20058

This theorem is referenced by: (None)