Theorem psrbagcon 14952

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 531	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ (𝑝 ∈ ℕ0 ∧ 𝑞 ∈ ℕ0)) → 𝑝 ∈ ℕ0)
2	1	nn0zd 9716	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ (𝑝 ∈ ℕ0 ∧ 𝑞 ∈ ℕ0)) → 𝑝 ∈ ℤ)
3		simprr 533	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ (𝑝 ∈ ℕ0 ∧ 𝑞 ∈ ℕ0)) → 𝑞 ∈ ℕ0)
4	3	nn0zd 9716	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ (𝑝 ∈ ℕ0 ∧ 𝑞 ∈ ℕ0)) → 𝑞 ∈ ℤ)
5	2, 4	zsubcld 9723	. . . . . 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 14944	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
8	7	3ad2ant1 1045	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹:𝐼⟶ℕ0)
9		simp2 1025	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺:𝐼⟶ℕ0)
10		id 19	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → 𝐹 ∈ 𝐷)
11	7	ffnd 5514	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
12	10, 11	fndmexd 5561	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 𝐼 ∈ V)
13	12	3ad2ant1 1045	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐼 ∈ V)
14		inidm 3434	. . . . . 6 ⊢ (𝐼 ∩ 𝐼) = 𝐼
15	5, 8, 9, 13, 13, 14	off 6288	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∘𝑓 − 𝐺):𝐼⟶ℤ)
16	15	ffnd 5514	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∘𝑓 − 𝐺) Fn 𝐼)
17	11	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹 Fn 𝐼)
18	9	ffnd 5514	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 Fn 𝐼)
19		eqidd 2235	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
20		eqidd 2235	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
21	8	ffvelcdmda 5817	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℕ0)
22	21	nn0zd 9716	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℤ)
23	9	ffvelcdmda 5817	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℕ0)
24	23	nn0zd 9716	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℤ)
25	22, 24	zsubcld 9723	. . . . . . 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 6285	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → ((𝐹 ∘𝑓 − 𝐺)‘𝑥) = ((𝐹‘𝑥) − (𝐺‘𝑥)))
27		simp3 1026	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 ∘𝑟 ≤ 𝐹)
28	18, 17, 13, 13, 14, 20, 19	ofrfval 6284	. . . . . . . . 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 2632	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
31		nn0sub 9661	. . . . . . . 8 ⊢ (((𝐺‘𝑥) ∈ ℕ0 ∧ (𝐹‘𝑥) ∈ ℕ0) → ((𝐺‘𝑥) ≤ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) − (𝐺‘𝑥)) ∈ ℕ0))
32	23, 21, 31	syl2anc 411	. . . . . . 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 2311	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → ((𝐹 ∘𝑓 − 𝐺)‘𝑥) ∈ ℕ0)
35	34	ralrimiva 2617	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ 𝐼 ((𝐹 ∘𝑓 − 𝐺)‘𝑥) ∈ ℕ0)
36		ffnfv 5840	. . . 4 ⊢ ((𝐹 ∘𝑓 − 𝐺):𝐼⟶ℕ0 ↔ ((𝐹 ∘𝑓 − 𝐺) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝐹 ∘𝑓 − 𝐺)‘𝑥) ∈ ℕ0))
37	16, 35, 36	sylanbrc 417	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∘𝑓 − 𝐺):𝐼⟶ℕ0)
38		simp1 1024	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹 ∈ 𝐷)
39	6	psrbag 14943	. . . . . . 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 9573	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → 0 ≤ (𝐺‘𝑥))
44	21	nn0red 9571	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℝ)
45	23	nn0red 9571	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℝ)
46	44, 45	subge02d 8828	. . . . . . . 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 2617	. . . . . 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 6284	. . . . . 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 14948	. . . . 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 1274	. . . 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 5802	. . . . . . . . . . . 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 5818	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℕ0)
61	60	nn0zd 9716	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑗 ∈ (◡𝐹 “ ℕ)) → ((𝐹 ∘𝑓 − 𝐺)‘𝑗) ∈ ℤ)
62		elnndc 9962	. . . . . . 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 5802	. . . . . . . . . 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 441	. . . . . . 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 846	. . . . . 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 2617	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ∀𝑗 ∈ (◡𝐹 “ ℕ)DECID 𝑗 ∈ (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ))
71		ssfidc 7211	. . . 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 1274	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡(𝐹 ∘𝑓 − 𝐺) “ ℕ) ∈ Fin)
73	6	psrbag 14943	. . . 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 953	. 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 842   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  {crab 2526  Vcvv 2815   ⊆ wss 3214   class class class wbr 4114  ^◡ccnv 4753   “ cima 4757   Fn wfn 5352  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058   ∘_𝑓 cof 6273   ∘_𝑟 cofr 6274   ↑_𝑚 cmap 6895  Fincfn 6988  0cc0 8143   ≤ cle 8325   − cmin 8460  ℕcn 9254  ℕ₀cn0 9513  ℤcz 9594

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-ofr 6276  df-1o 6660  df-er 6780  df-map 6897  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872

This theorem is referenced by:  psrbagconcl  14953