Theorem mplsubrglemcl 15249

Description: Lemma for mplsubrgfi 15250. Polynomials are closed under multiplication. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by AV, 18-Jul-2019.)

Hypotheses

Ref	Expression
mplsubg.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
mplsubg.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplsubg.u	⊢ 𝑈 = (Base‘𝑃)
mplsubg.i	⊢ (𝜑 → 𝐼 ∈ Fin)
mpllss.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mplsubrglem.x	⊢ (𝜑 → 𝑋 ∈ 𝑈)
mplsubrglem.y	⊢ (𝜑 → 𝑌 ∈ 𝑈)

Assertion

Ref	Expression
mplsubrglemcl	⊢ (𝜑 → (𝑋(.r‘𝑆)𝑌) ∈ 𝑈)

Proof of Theorem mplsubrglemcl

Dummy variables 𝑘 𝑠 𝑝 𝑡 𝑥 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplsubg.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		eqid 2238	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2238	. . 3 ⊢ (.r‘𝑆) = (.r‘𝑆)
4		mpllss.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
5		mplsubg.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ Fin)
6		mplsubg.p	. . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
7		mplsubg.u	. . . . 5 ⊢ 𝑈 = (Base‘𝑃)
8	6, 1, 7, 2	mplbasss 15243	. . . 4 ⊢ 𝑈 ⊆ (Base‘𝑆)
9		mplsubrglem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
10	8, 9	sselid 3246	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑆))
11		mplsubrglem.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
12	8, 11	sselid 3246	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑆))
13	1, 2, 3, 4, 5, 10, 12	psrmulclfi 15210	. 2 ⊢ (𝜑 → (𝑋(.r‘𝑆)𝑌) ∈ (Base‘𝑆))
14		eqid 2238	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	1, 14, 5, 2, 10	psrelbasfi 15199	. . . 4 ⊢ (𝜑 → 𝑋:(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
16	4	ringgrpd 14416	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
17		eqid 2238	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
18	14, 17	grpidcl 13893	. . . . 5 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ (Base‘𝑅))
19	16, 18	syl 14	. . . 4 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
20	6, 7, 17, 9	mplelsfi 15241	. . . 4 ⊢ (𝜑 → 𝑋 finSupp (0g‘𝑅))
21	15, 19, 5, 20	psrbagfsuppub 15192	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))
22	1, 14, 5, 2, 12	psrelbasfi 15199	. . . . . 6 ⊢ (𝜑 → 𝑌:(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
23	6, 7, 17, 11	mplelsfi 15241	. . . . . 6 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅))
24	22, 19, 5, 23	psrbagfsuppub 15192	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))
25	24	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) → ∃𝑡 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))
26	1, 14, 5, 2, 13	psrelbasfi 15199	. . . . . 6 ⊢ (𝜑 → (𝑋(.r‘𝑆)𝑌):(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
27	26	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → (𝑋(.r‘𝑆)𝑌):(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
28	19	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → (0g‘𝑅) ∈ (Base‘𝑅))
29	5	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → 𝐼 ∈ Fin)
30		nn0addcl 9610	. . . . . . . 8 ⊢ ((𝑢 ∈ ℕ0 ∧ 𝑣 ∈ ℕ0) → (𝑢 + 𝑣) ∈ ℕ0)
31	30	adantl 277	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ (𝑢 ∈ ℕ0 ∧ 𝑣 ∈ ℕ0)) → (𝑢 + 𝑣) ∈ ℕ0)
32		simplrl 541	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → 𝑠 ∈ (ℕ0 ↑𝑚 𝐼))
33		elmapi 6949	. . . . . . . 8 ⊢ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) → 𝑠:𝐼⟶ℕ0)
34	32, 33	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → 𝑠:𝐼⟶ℕ0)
35		simprl 535	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → 𝑡 ∈ (ℕ0 ↑𝑚 𝐼))
36		elmapi 6949	. . . . . . . 8 ⊢ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) → 𝑡:𝐼⟶ℕ0)
37	35, 36	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → 𝑡:𝐼⟶ℕ0)
38		inidm 3440	. . . . . . 7 ⊢ (𝐼 ∩ 𝐼) = 𝐼
39	31, 34, 37, 29, 29, 38	off 6319	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → (𝑠 ∘𝑓 + 𝑡):𝐼⟶ℕ0)
40		nn0ex 9581	. . . . . . . 8 ⊢ ℕ0 ∈ V
41	40	a1i 9	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → ℕ0 ∈ V)
42	41, 29	elmapd 6941	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → ((𝑠 ∘𝑓 + 𝑡) ∈ (ℕ0 ↑𝑚 𝐼) ↔ (𝑠 ∘𝑓 + 𝑡):𝐼⟶ℕ0))
43	39, 42	mpbird 167	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → (𝑠 ∘𝑓 + 𝑡) ∈ (ℕ0 ↑𝑚 𝐼))
44		eqid 2238	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
45	10	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑋 ∈ (Base‘𝑆))
46	12	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑌 ∈ (Base‘𝑆))
47		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑘 ∈ (ℕ0 ↑𝑚 𝐼))
48	5	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝐼 ∈ Fin)
49	4	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑅 ∈ Ring)
50	1, 2, 44, 3, 45, 46, 47, 48, 49	psrmulvalfifi 15208	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → ((𝑋(.r‘𝑆)𝑌)‘𝑘) = (𝑅 Σg (𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))))
51	50	ad5ant14 524	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) → ((𝑋(.r‘𝑆)𝑌)‘𝑘) = (𝑅 Σg (𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))))
52		breq1 4134	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑥 → (𝑝 ∘𝑟 ≤ 𝑠 ↔ 𝑥 ∘𝑟 ≤ 𝑠))
53	52	notbid 677	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑥 → (¬ 𝑝 ∘𝑟 ≤ 𝑠 ↔ ¬ 𝑥 ∘𝑟 ≤ 𝑠))
54		fveqeq2 5707	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑥 → ((𝑋‘𝑝) = (0g‘𝑅) ↔ (𝑋‘𝑥) = (0g‘𝑅)))
55	53, 54	imbi12d 234	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑥 → ((¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)) ↔ (¬ 𝑥 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑥) = (0g‘𝑅))))
56		simplrr 542	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))
57	56	ad4antr 498	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))
58		elrabi 2979	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} → 𝑥 ∈ (ℕ0 ↑𝑚 𝐼))
59	58	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑥 ∈ (ℕ0 ↑𝑚 𝐼))
60	59	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → 𝑥 ∈ (ℕ0 ↑𝑚 𝐼))
61	55, 57, 60	rspcdva 2934	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → (¬ 𝑥 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑥) = (0g‘𝑅)))
62	61	syldbl2 1333	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → (𝑋‘𝑥) = (0g‘𝑅))
63	62	oveq1d 6103	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = ((0g‘𝑅)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))
64	4	ad6antr 502	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → 𝑅 ∈ Ring)
65	22	ad6antr 502	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → 𝑌:(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
66		simpllr 540	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ (ℕ0 ↑𝑚 𝐼))
67		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘})
68	29	ad3antrrr 496	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ Fin)
69		eqid 2238	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} = {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}
70	69	psrbagconclfi 15182	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (ℕ0 ↑𝑚 𝐼) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ∧ 𝐼 ∈ Fin) → (𝑘 ∘𝑓 − 𝑥) ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘})
71		elrabi 2979	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∘𝑓 − 𝑥) ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} → (𝑘 ∘𝑓 − 𝑥) ∈ (ℕ0 ↑𝑚 𝐼))
72	70, 71	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (ℕ0 ↑𝑚 𝐼) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ∧ 𝐼 ∈ Fin) → (𝑘 ∘𝑓 − 𝑥) ∈ (ℕ0 ↑𝑚 𝐼))
73	66, 67, 68, 72	syl3anc 1278	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑥) ∈ (ℕ0 ↑𝑚 𝐼))
74	73	adantr 276	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → (𝑘 ∘𝑓 − 𝑥) ∈ (ℕ0 ↑𝑚 𝐼))
75	65, 74	ffvelcdmd 5847	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → (𝑌‘(𝑘 ∘𝑓 − 𝑥)) ∈ (Base‘𝑅))
76	14, 44, 17, 64, 75	ringlzd 14457	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → ((0g‘𝑅)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = (0g‘𝑅))
77	63, 76	eqtrd 2271	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ 𝑥 ∘𝑟 ≤ 𝑠) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = (0g‘𝑅))
78		breq1 4134	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑘 ∘𝑓 − 𝑥) → (𝑝 ∘𝑟 ≤ 𝑡 ↔ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡))
79	78	notbid 677	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑘 ∘𝑓 − 𝑥) → (¬ 𝑝 ∘𝑟 ≤ 𝑡 ↔ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡))
80		fveqeq2 5707	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (𝑘 ∘𝑓 − 𝑥) → ((𝑌‘𝑝) = (0g‘𝑅) ↔ (𝑌‘(𝑘 ∘𝑓 − 𝑥)) = (0g‘𝑅)))
81	79, 80	imbi12d 234	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (𝑘 ∘𝑓 − 𝑥) → ((¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)) ↔ (¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡 → (𝑌‘(𝑘 ∘𝑓 − 𝑥)) = (0g‘𝑅))))
82		simprr 537	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))
83	82	ad4antr 498	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))
84	73	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → (𝑘 ∘𝑓 − 𝑥) ∈ (ℕ0 ↑𝑚 𝐼))
85	81, 83, 84	rspcdva 2934	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → (¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡 → (𝑌‘(𝑘 ∘𝑓 − 𝑥)) = (0g‘𝑅)))
86	85	syldbl2 1333	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → (𝑌‘(𝑘 ∘𝑓 − 𝑥)) = (0g‘𝑅))
87	86	oveq2d 6104	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = ((𝑋‘𝑥)(.r‘𝑅)(0g‘𝑅)))
88	4	ad6antr 502	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → 𝑅 ∈ Ring)
89	15	ad6antr 502	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → 𝑋:(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
90	59	adantr 276	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → 𝑥 ∈ (ℕ0 ↑𝑚 𝐼))
91	89, 90	ffvelcdmd 5847	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → (𝑋‘𝑥) ∈ (Base‘𝑅))
92	14, 44, 17, 88, 91	ringrzd 14458	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → ((𝑋‘𝑥)(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
93	87, 92	eqtrd 2271	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = (0g‘𝑅))
94		elmapi 6949	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℕ0 ↑𝑚 𝐼) → 𝑥:𝐼⟶ℕ0)
95	59, 94	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑥:𝐼⟶ℕ0)
96	95	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑥:𝐼⟶ℕ0)
97		nn0sscn 9580	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ⊆ ℂ
98		fss 5549	. . . . . . . . . . . . . . 15 ⊢ ((𝑥:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑥:𝐼⟶ℂ)
99	96, 97, 98	sylancl 417	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑥:𝐼⟶ℂ)
100		elmapi 6949	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℕ0 ↑𝑚 𝐼) → 𝑘:𝐼⟶ℕ0)
101	100	ad4antlr 499	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑘:𝐼⟶ℕ0)
102		fss 5549	. . . . . . . . . . . . . . 15 ⊢ ((𝑘:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑘:𝐼⟶ℂ)
103	101, 97, 102	sylancl 417	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑘:𝐼⟶ℂ)
104	68	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝐼 ∈ Fin)
105	99, 103, 104	ofpncan3 15181	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → (𝑥 ∘𝑓 + (𝑘 ∘𝑓 − 𝑥)) = 𝑘)
106		nn0ssre 9579	. . . . . . . . . . . . . . 15 ⊢ ℕ0 ⊆ ℝ
107		fss 5549	. . . . . . . . . . . . . . 15 ⊢ ((𝑥:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℝ) → 𝑥:𝐼⟶ℝ)
108	96, 106, 107	sylancl 417	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑥:𝐼⟶ℝ)
109		elmapi 6949	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∘𝑓 − 𝑥) ∈ (ℕ0 ↑𝑚 𝐼) → (𝑘 ∘𝑓 − 𝑥):𝐼⟶ℕ0)
110	73, 109	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑥):𝐼⟶ℕ0)
111	110	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → (𝑘 ∘𝑓 − 𝑥):𝐼⟶ℕ0)
112		fss 5549	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∘𝑓 − 𝑥):𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℝ) → (𝑘 ∘𝑓 − 𝑥):𝐼⟶ℝ)
113	111, 106, 112	sylancl 417	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → (𝑘 ∘𝑓 − 𝑥):𝐼⟶ℝ)
114	34	ad4antr 498	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑠:𝐼⟶ℕ0)
115		fss 5549	. . . . . . . . . . . . . . 15 ⊢ ((𝑠:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℝ) → 𝑠:𝐼⟶ℝ)
116	114, 106, 115	sylancl 417	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑠:𝐼⟶ℝ)
117	37	ad4antr 498	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑡:𝐼⟶ℕ0)
118		fss 5549	. . . . . . . . . . . . . . 15 ⊢ ((𝑡:𝐼⟶ℕ0 ∧ ℕ0 ⊆ ℝ) → 𝑡:𝐼⟶ℝ)
119	117, 106, 118	sylancl 417	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑡:𝐼⟶ℝ)
120		simprl 535	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑥 ∘𝑟 ≤ 𝑠)
121		simprr 537	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)
122	108, 113, 116, 119, 104, 120, 121	ofrle2add 15180	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → (𝑥 ∘𝑓 + (𝑘 ∘𝑓 − 𝑥)) ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡))
123	105, 122	eqbrtrrd 4155	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡))
124		simpllr 540	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) ∧ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)) → ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡))
125	123, 124	pm2.65da 671	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → ¬ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡))
126	32	ad3antrrr 496	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → 𝑠 ∈ (ℕ0 ↑𝑚 𝐼))
127		psrbagdcle 15191	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∧ 𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ 𝐼 ∈ Fin) → DECID 𝑥 ∘𝑟 ≤ 𝑠)
128	59, 126, 68, 127	syl3anc 1278	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → DECID 𝑥 ∘𝑟 ≤ 𝑠)
129		ianordc 911	. . . . . . . . . . . 12 ⊢ (DECID 𝑥 ∘𝑟 ≤ 𝑠 → (¬ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) ↔ (¬ 𝑥 ∘𝑟 ≤ 𝑠 ∨ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)))
130	128, 129	syl 14	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (¬ (𝑥 ∘𝑟 ≤ 𝑠 ∧ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡) ↔ (¬ 𝑥 ∘𝑟 ≤ 𝑠 ∨ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡)))
131	125, 130	mpbid 147	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → (¬ 𝑥 ∘𝑟 ≤ 𝑠 ∨ ¬ (𝑘 ∘𝑓 − 𝑥) ∘𝑟 ≤ 𝑡))
132	77, 93, 131	mpjaodan 810	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) ∧ 𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘}) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))) = (0g‘𝑅))
133	132	mpteq2dva 4223	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) → (𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥)))) = (𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (0g‘𝑅)))
134	133	oveq2d 6104	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘𝑓 − 𝑥))))) = (𝑅 Σg (𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (0g‘𝑅))))
135	4	ringcmnd 14447	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CMnd)
136	135	ad4antr 498	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) → 𝑅 ∈ CMnd)
137		psrbaglefimap 15183	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℕ0 ↑𝑚 𝐼) ∧ 𝐼 ∈ Fin) → {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ∈ Fin)
138	47, 48, 137	syl2anc 415	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ∈ Fin)
139	138	ad5ant14 524	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) → {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ∈ Fin)
140	17	gsumzfi 14248	. . . . . . . 8 ⊢ ((𝑅 ∈ CMnd ∧ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ∈ Fin) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (0g‘𝑅))) = (0g‘𝑅))
141	136, 139, 140	syl2anc 415	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ (0g‘𝑅))) = (0g‘𝑅))
142	51, 134, 141	3eqtrd 2275	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡)) → ((𝑋(.r‘𝑆)𝑌)‘𝑘) = (0g‘𝑅))
143	142	ex 115	. . . . 5 ⊢ ((((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → (¬ 𝑘 ∘𝑟 ≤ (𝑠 ∘𝑓 + 𝑡) → ((𝑋(.r‘𝑆)𝑌)‘𝑘) = (0g‘𝑅)))
144	27, 28, 29, 43, 143	psrbagfsuppd 15193	. . . 4 ⊢ (((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) ∧ (𝑡 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑡 → (𝑌‘𝑝) = (0g‘𝑅)))) → (𝑋(.r‘𝑆)𝑌) finSupp (0g‘𝑅))
145	25, 144	rexlimddv 2673	. . 3 ⊢ ((𝜑 ∧ (𝑠 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑝 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑝 ∘𝑟 ≤ 𝑠 → (𝑋‘𝑝) = (0g‘𝑅)))) → (𝑋(.r‘𝑆)𝑌) finSupp (0g‘𝑅))
146	21, 145	rexlimddv 2673	. 2 ⊢ (𝜑 → (𝑋(.r‘𝑆)𝑌) finSupp (0g‘𝑅))
147	6, 1, 2, 17, 7	mplelbas 15236	. 2 ⊢ ((𝑋(.r‘𝑆)𝑌) ∈ 𝑈 ↔ ((𝑋(.r‘𝑆)𝑌) ∈ (Base‘𝑆) ∧ (𝑋(.r‘𝑆)𝑌) finSupp (0g‘𝑅)))
148	13, 146, 147	sylanbrc 421	1 ⊢ (𝜑 → (𝑋(.r‘𝑆)𝑌) ∈ 𝑈)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821   ⊆ wss 3220   class class class wbr 4131   ↦ cmpt 4193  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088   ∘_𝑓 cof 6303   ∘_𝑟 cofr 6304   ↑_𝑚 cmap 6927  Fincfn 7028   finSupp cfsupp 7292  ℂcc 8184  ℝcr 8185   + caddc 8189   ≤ cle 8368   − cmin 8505  ℕ₀cn0 9575  Basecbs 13410  ._rcmulr 13491  0_gc0g 13669  Grpcgrp 13864  CMndccmn 14177   Σ_g cgsu 14240  Ringcrg 14407   mPwSer cmps 15154   mPoly cmpl 15156

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-stab 843  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-rmo 2536  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 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-ofr 6306  df-1st 6378  df-2nd 6379  df-recs 6581  df-irdg 6646  df-frec 6667  df-1o 6692  df-oadd 6696  df-er 6812  df-map 6929  df-ixp 6987  df-en 7029  df-dom 7030  df-fin 7031  df-fsuppfi 7293  df-sup 7331  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-5 9376  df-6 9377  df-7 9378  df-8 9379  df-9 9380  df-n0 9576  df-z 9655  df-uz 9938  df-rp 10072  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-struct 13412  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-mulr 13504  df-sca 13506  df-vsca 13507  df-tset 13509  df-rest 13654  df-topn 13655  df-0g 13671  df-gzsum 13672  df-topgen 13673  df-pt 13674  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-mulg 13982  df-cmn 14179  df-abl 14180  df-gsumfi 14241  df-mgp 14325  df-ur 14370  df-ring 14409  df-psr 15157  df-mpl 15159

This theorem is used by:  mplsubrgfi  15250