Theorem psrbagcon 15045

Description: The analogue of the statement "0 ≤ 𝐺 ≤ 𝐹 implies 0 ≤ 𝐹 − 𝐺 ≤ 𝐹 " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)

Hypothesis

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

Assertion

Ref	Expression
psrbagcon	⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ((𝐹 ∘𝑓 − 𝐺) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝐺) ∘𝑟 ≤ 𝐹))

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

Allowed substitution hint:   𝐷(𝑓)

Proof of Theorem psrbagcon

Dummy variables 𝑥 𝑗 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 535	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ (𝑝 ∈ ℕ0 ∧ 𝑞 ∈ ℕ0)) → 𝑝 ∈ ℕ0)
2	1	nn0zd 9749	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ (𝑝 ∈ ℕ0 ∧ 𝑞 ∈ ℕ0)) → 𝑝 ∈ ℤ)
3		simprr 537	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ (𝑝 ∈ ℕ0 ∧ 𝑞 ∈ ℕ0)) → 𝑞 ∈ ℕ0)
4	3	nn0zd 9749	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ (𝑝 ∈ ℕ0 ∧ 𝑞 ∈ ℕ0)) → 𝑞 ∈ ℤ)
5	2, 4	zsubcld 9756	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ (𝑝 ∈ ℕ0 ∧ 𝑞 ∈ ℕ0)) → (𝑝 − 𝑞) ∈ ℤ)
6		psrbag.d	. . . . . . . 8 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
7	6	psrbagf 15037	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
8	7	3ad2ant1 1049	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹:𝐼⟶ℕ0)
9		simp2 1029	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺:𝐼⟶ℕ0)
10		id 19	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → 𝐹 ∈ 𝐷)
11	7	ffnd 5532	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
12	10, 11	fndmexd 5579	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 𝐼 ∈ V)
13	12	3ad2ant1 1049	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐼 ∈ V)
14		inidm 3440	. . . . . 6 ⊢ (𝐼 ∩ 𝐼) = 𝐼
15	5, 8, 9, 13, 13, 14	off 6309	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∘𝑓 − 𝐺):𝐼⟶ℤ)
16	15	ffnd 5532	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∘𝑓 − 𝐺) Fn 𝐼)
17	11	3ad2ant1 1049	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹 Fn 𝐼)
18	9	ffnd 5532	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 Fn 𝐼)
19		eqidd 2239	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
20		eqidd 2239	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
21	8	ffvelcdmda 5837	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℕ0)
22	21	nn0zd 9749	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℤ)
23	9	ffvelcdmda 5837	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℕ0)
24	23	nn0zd 9749	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℤ)
25	22, 24	zsubcld 9756	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) − (𝐺‘𝑥)) ∈ ℤ)
26	17, 18, 13, 13, 14, 19, 20, 25	ofvalg 6306	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → ((𝐹 ∘𝑓 − 𝐺)‘𝑥) = ((𝐹‘𝑥) − (𝐺‘𝑥)))
27		simp3 1030	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 ∘𝑟 ≤ 𝐹)
28	18, 17, 13, 13, 14, 20, 19	ofrfval 6305	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐺 ∘𝑟 ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥)))
29	27, 28	mpbid 147	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥))
30	29	r19.21bi 2638	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
31		nn0sub 9694	. . . . . . . 8 ⊢ (((𝐺‘𝑥) ∈ ℕ0 ∧ (𝐹‘𝑥) ∈ ℕ0) → ((𝐺‘𝑥) ≤ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) − (𝐺‘𝑥)) ∈ ℕ0))
32	23, 21, 31	syl2anc 415	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → ((𝐺‘𝑥) ≤ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) − (𝐺‘𝑥)) ∈ ℕ0))
33	30, 32	mpbid 147	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) − (𝐺‘𝑥)) ∈ ℕ0)
34	26, 33	eqeltrd 2315	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → ((𝐹 ∘𝑓 − 𝐺)‘𝑥) ∈ ℕ0)
35	34	ralrimiva 2623	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ 𝐼 ((𝐹 ∘𝑓 − 𝐺)‘𝑥) ∈ ℕ0)
36		ffnfv 5860	. . . 4 ⊢ ((𝐹 ∘𝑓 − 𝐺):𝐼⟶ℕ0 ↔ ((𝐹 ∘𝑓 − 𝐺) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝐹 ∘𝑓 − 𝐺)‘𝑥) ∈ ℕ0))
37	16, 35, 36	sylanbrc 421	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∘𝑓 − 𝐺):𝐼⟶ℕ0)
38		simp1 1028	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹 ∈ 𝐷)
39	6	psrbag 15036	. . . . . . 7 ⊢ (𝐼 ∈ V → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)))
40	13, 39	syl 14	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)))
41	38, 40	mpbid 147	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))
42	41	simprd 114	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡𝐹 “ ℕ) ∈ Fin)
43	23	nn0ge0d 9606	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → 0 ≤ (𝐺‘𝑥))
44	21	nn0red 9604	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℝ)
45	23	nn0red 9604	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℝ)
46	44, 45	subge02d 8859	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (0 ≤ (𝐺‘𝑥) ↔ ((𝐹‘𝑥) − (𝐺‘𝑥)) ≤ (𝐹‘𝑥)))
47	43, 46	mpbid 147	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) − (𝐺‘𝑥)) ≤ (𝐹‘𝑥))
48	47	ralrimiva 2623	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥) − (𝐺‘𝑥)) ≤ (𝐹‘𝑥))
49	16, 17, 13, 13, 14, 26, 19	ofrfval 6305	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ((𝐹 ∘𝑓 − 𝐺) ∘𝑟 ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥) − (𝐺‘𝑥)) ≤ (𝐹‘𝑥)))
50	48, 49	mpbird 167	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∘𝑓 − 𝐺) ∘𝑟 ≤ 𝐹)
51	6	psrbaglesupp 15041	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝐺):𝐼⟶ℕ0 ∧ (𝐹 ∘𝑓 − 𝐺) ∘𝑟 ≤ 𝐹) → (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ⊆ (◡𝐹 “ ℕ))
52	38, 37, 50, 51	syl3anc 1278	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ⊆ (◡𝐹 “ ℕ))
53	37	adantr 276	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → (𝐹 ∘𝑓 − 𝐺):𝐼⟶ℕ0)
54		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → 𝑗 ∈ (◡𝐹 “ ℕ))
55	17	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → 𝐹 Fn 𝐼)
56		elpreima 5822	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐼 → (𝑗 ∈ (◡𝐹 “ ℕ) ↔ (𝑗 ∈ 𝐼 ∧ (𝐹‘𝑗) ∈ ℕ)))
57	55, 56	syl 14	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → (𝑗 ∈ (◡𝐹 “ ℕ) ↔ (𝑗 ∈ 𝐼 ∧ (𝐹‘𝑗) ∈ ℕ)))
58	54, 57	mpbid 147	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → (𝑗 ∈ 𝐼 ∧ (𝐹‘𝑗) ∈ ℕ))
59	58	simpld 112	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → 𝑗 ∈ 𝐼)
60	53, 59	ffvelcdmd 5838	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℕ0)
61	60	nn0zd 9749	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℤ)
62		elnndc 9995	. . . . . . 7 ⊢ (((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℤ → DECID ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℕ)
63	61, 62	syl 14	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → DECID ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℕ)
64		elpreima 5822	. . . . . . . . . 10 ⊢ ((𝐹 ∘𝑓 − 𝐺) Fn 𝐼 → (𝑗 ∈ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ↔ (𝑗 ∈ 𝐼 ∧ ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℕ)))
65	16, 64	syl 14	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝑗 ∈ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ↔ (𝑗 ∈ 𝐼 ∧ ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℕ)))
66	65	adantr 276	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → (𝑗 ∈ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ↔ (𝑗 ∈ 𝐼 ∧ ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℕ)))
67	59, 66	mpbirand 445	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → (𝑗 ∈ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ↔ ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℕ))
68	67	dcbid 850	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → (DECID 𝑗 ∈ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ↔ DECID ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℕ))
69	63, 68	mpbird 167	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → DECID 𝑗 ∈ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ))
70	69	ralrimiva 2623	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ∀𝑗 ∈ (◡𝐹 “ ℕ)DECID 𝑗 ∈ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ))
71		ssfidc 7239	. . . 4 ⊢ (((◡𝐹 “ ℕ) ∈ Fin ∧ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ⊆ (◡𝐹 “ ℕ) ∧ ∀𝑗 ∈ (◡𝐹 “ ℕ)DECID 𝑗 ∈ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ)) → (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ∈ Fin)
72	42, 52, 70, 71	syl3anc 1278	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ∈ Fin)
73	6	psrbag 15036	. . . 4 ⊢ (𝐼 ∈ V → ((𝐹 ∘𝑓 − 𝐺) ∈ 𝐷 ↔ ((𝐹 ∘𝑓 − 𝐺):𝐼⟶ℕ0 ∧ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ∈ Fin)))
74	13, 73	syl 14	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ((𝐹 ∘𝑓 − 𝐺) ∈ 𝐷 ↔ ((𝐹 ∘𝑓 − 𝐺):𝐼⟶ℕ0 ∧ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ∈ Fin)))
75	37, 72, 74	mpbir2and 957	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∘𝑓 − 𝐺) ∈ 𝐷)
76	75, 50	jca 306	1 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ((𝐹 ∘𝑓 − 𝐺) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝐺) ∘𝑟 ≤ 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ⊆ wss 3220   class class class wbr 4128  ^◡ccnv 4771   “ cima 4775   Fn wfn 5370  ⟶wf 5371  ‘cfv 5375  (class class class)co 6079   ∘_𝑓 cof 6294   ∘_𝑟 cofr 6295   ↑_𝑚 cmap 6916  Fincfn 7016  0cc0 8173   ≤ cle 8355   − cmin 8491  ℕcn 9287  ℕ₀cn0 9546  ℤcz 9627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296  df-ofr 6297  df-1o 6681  df-er 6801  df-map 6918  df-en 7017  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905

This theorem is referenced by:  psrbagconcl  15046