Theorem rrxsuppss 23186

Description: Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.)

Hypotheses

Ref	Expression
rrxmval.1	⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}
rrxf.1	⊢ (𝜑 → 𝐹 ∈ 𝑋)

Assertion

Ref	Expression
rrxsuppss	⊢ (𝜑 → (𝐹 supp 0) ⊆ 𝐼)

Distinct variable groups:   ℎ,𝐹   ℎ,𝐼

Allowed substitution hints:   𝜑(ℎ)   𝑋(ℎ)

Proof of Theorem rrxsuppss

Step	Hyp	Ref	Expression
1		suppssdm 7308	. 2 ⊢ (𝐹 supp 0) ⊆ dom 𝐹
2		rrxmval.1	. . . 4 ⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}
3		rrxf.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
4	2, 3	rrxf 23184	. . 3 ⊢ (𝜑 → 𝐹:𝐼⟶ℝ)
5		fdm 6051	. . 3 ⊢ (𝐹:𝐼⟶ℝ → dom 𝐹 = 𝐼)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐼)
7	1, 6	syl5sseq 3653	1 ⊢ (𝜑 → (𝐹 supp 0) ⊆ 𝐼)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {crab 2916   ⊆ wss 3574   class class class wbr 4653  dom cdm 5114  ⟶wf 5884  (class class class)co 6650   supp csupp 7295   ↑_𝑚 cmap 7857   finSupp cfsupp 8275  ℝcr 9935  0cc0 9936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-supp 7296  df-map 7859

This theorem is referenced by:  rrxmval  23188  rrxmet  23191  rrxdstprj1  23192