Theorem psrbaglesuppOLD 20701

Description: Obsolete version of psrbaglesupp 20700 as of 5-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
psrbag.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}

Assertion

Ref	Expression
psrbaglesuppOLD	⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Distinct variable groups:   𝑓,𝐹   𝑓,𝐺   𝑓,𝐼

Allowed substitution hints:   𝐷(𝑓)   𝑉(𝑓)

Proof of Theorem psrbaglesuppOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frnnn0supp 12003	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺:𝐼⟶ℕ0) → (𝐺 supp 0) = (◡𝐺 “ ℕ))
2	1	3ad2antr2 1186	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → (𝐺 supp 0) = (◡𝐺 “ ℕ))
3		simpr2 1192	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → 𝐺:𝐼⟶ℕ0)
4		eldifi 4034	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ)) → 𝑥 ∈ 𝐼)
5		simpr3 1193	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → 𝐺 ∘r ≤ 𝐹)
6	3	ffnd 6504	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → 𝐺 Fn 𝐼)
7		psrbag.d	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
8	7	psrbagfOLD 20695	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0)
9	8	3ad2antr1 1185	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → 𝐹:𝐼⟶ℕ0)
10	9	ffnd 6504	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → 𝐹 Fn 𝐼)
11		simpl 486	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → 𝐼 ∈ 𝑉)
12		inidm 4125	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
13		eqidd 2759	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
14		eqidd 2759	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
15	6, 10, 11, 11, 12, 13, 14	ofrfval 7420	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → (𝐺 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥)))
16	5, 15	mpbid 235	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥))
17	16	r19.21bi 3137	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
18	4, 17	sylan2 595	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
19	11, 9	jca 515	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → (𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0))
20		frnnn0supp 12003	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
21		eqimss 3950	. . . . . . 7 ⊢ ((𝐹 supp 0) = (◡𝐹 “ ℕ) → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
22	19, 20, 21	3syl 18	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
23		c0ex 10686	. . . . . . 7 ⊢ 0 ∈ V
24	23	a1i 11	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → 0 ∈ V)
25	9, 22, 11, 24	suppssr 7876	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
26	18, 25	breqtrd 5062	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ≤ 0)
27		ffvelrn 6846	. . . . . 6 ⊢ ((𝐺:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℕ0)
28	3, 4, 27	syl2an 598	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ∈ ℕ0)
29	28	nn0ge0d 12010	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → 0 ≤ (𝐺‘𝑥))
30	28	nn0red 12008	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ∈ ℝ)
31		0re 10694	. . . . 5 ⊢ 0 ∈ ℝ
32		letri3 10777	. . . . 5 ⊢ (((𝐺‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝐺‘𝑥) = 0 ↔ ((𝐺‘𝑥) ≤ 0 ∧ 0 ≤ (𝐺‘𝑥))))
33	30, 31, 32	sylancl 589	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → ((𝐺‘𝑥) = 0 ↔ ((𝐺‘𝑥) ≤ 0 ∧ 0 ≤ (𝐺‘𝑥))))
34	26, 29, 33	mpbir2and 712	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) = 0)
35	3, 34	suppss 7874	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → (𝐺 supp 0) ⊆ (◡𝐹 “ ℕ))
36	2, 35	eqsstrrd 3933	1 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  Vcvv 3409   ∖ cdif 3857   ⊆ wss 3860   class class class wbr 5036  ^◡ccnv 5527   “ cima 5531  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ∘_r cofr 7410   supp csupp 7841   ↑_m cmap 8422  Fincfn 8540  ℝcr 10587  0cc0 10588   ≤ cle 10727  ℕcn 11687  ℕ₀cn0 11947

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 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-ofr 7412  df-om 7586  df-supp 7842  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-map 8424  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-n0 11948

This theorem is referenced by:  psrbagleclOLD  20703  psrbagconOLD  20707