Theorem mplsubgfilemcl 14983

Description: Lemma for mplsubgfi 14985. The sum of two polynomials is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)

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)
mplsubg.r	⊢ (𝜑 → 𝑅 ∈ Grp)
mplsubgfilemcl.x	⊢ (𝜑 → 𝑋 ∈ 𝑈)
mplsubgfilemcl.y	⊢ (𝜑 → 𝑌 ∈ 𝑈)
mplsubgfilemcl.p	⊢ + = (+g‘𝑆)

Assertion

Ref	Expression
mplsubgfilemcl	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)

Proof of Theorem mplsubgfilemcl

Dummy variables 𝑎 𝑏 𝑝 𝑞 𝑘 𝑐 𝑑 𝑢 𝑣 𝑒 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplsubg.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		eqid 2234	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		mplsubgfilemcl.p	. . 3 ⊢ + = (+g‘𝑆)
4		mplsubg.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
5	4	grpmgmd 13784	. . 3 ⊢ (𝜑 → 𝑅 ∈ Mgm)
6		mplsubg.p	. . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
7		mplsubg.u	. . . . 5 ⊢ 𝑈 = (Base‘𝑃)
8	6, 1, 7, 2	mplbasss 14980	. . . 4 ⊢ 𝑈 ⊆ (Base‘𝑆)
9		mplsubgfilemcl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
10	8, 9	sselid 3240	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑆))
11		mplsubgfilemcl.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
12	8, 11	sselid 3240	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑆))
13	1, 2, 3, 5, 10, 12	psraddcl 14964	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (Base‘𝑆))
14		mplsubg.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ Fin)
15		eqid 2234	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
16	6, 1, 2, 15, 7	mplelbascoe 14976	. . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Grp) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ (Base‘𝑆) ∧ ∃𝑝 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))))
17	14, 4, 16	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ (Base‘𝑆) ∧ ∃𝑝 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))))
18	9, 17	mpbid 147	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝑆) ∧ ∃𝑝 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅))))
19	18	simprd 114	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))
20	6, 1, 2, 15, 7	mplelbascoe 14976	. . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Grp) → (𝑌 ∈ 𝑈 ↔ (𝑌 ∈ (Base‘𝑆) ∧ ∃𝑞 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))))
21	14, 4, 20	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ 𝑈 ↔ (𝑌 ∈ (Base‘𝑆) ∧ ∃𝑞 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))))
22	11, 21	mpbid 147	. . . . . 6 ⊢ (𝜑 → (𝑌 ∈ (Base‘𝑆) ∧ ∃𝑞 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅))))
23	22	simprd 114	. . . . 5 ⊢ (𝜑 → ∃𝑞 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))
24	23	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) → ∃𝑞 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))
25		nn0addcl 9551	. . . . . . . 8 ⊢ ((𝑐 ∈ ℕ0 ∧ 𝑑 ∈ ℕ0) → (𝑐 + 𝑑) ∈ ℕ0)
26	25	adantl 277	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ (𝑐 ∈ ℕ0 ∧ 𝑑 ∈ ℕ0)) → (𝑐 + 𝑑) ∈ ℕ0)
27		simplrl 537	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → 𝑝 ∈ (ℕ0 ↑𝑚 𝐼))
28		nn0ex 9522	. . . . . . . . . . 11 ⊢ ℕ0 ∈ V
29	28	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → ℕ0 ∈ V)
30	29, 14	elmapd 6909	. . . . . . . . 9 ⊢ (𝜑 → (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ↔ 𝑝:𝐼⟶ℕ0))
31	30	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ↔ 𝑝:𝐼⟶ℕ0))
32	27, 31	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → 𝑝:𝐼⟶ℕ0)
33		simprl 531	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → 𝑞 ∈ (ℕ0 ↑𝑚 𝐼))
34	29, 14	elmapd 6909	. . . . . . . . 9 ⊢ (𝜑 → (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ↔ 𝑞:𝐼⟶ℕ0))
35	34	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ↔ 𝑞:𝐼⟶ℕ0))
36	33, 35	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → 𝑞:𝐼⟶ℕ0)
37	14	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → 𝐼 ∈ Fin)
38		inidm 3434	. . . . . . 7 ⊢ (𝐼 ∩ 𝐼) = 𝐼
39	26, 32, 36, 37, 37, 38	off 6288	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → (𝑝 ∘𝑓 + 𝑞):𝐼⟶ℕ0)
40	29, 14	elmapd 6909	. . . . . . 7 ⊢ (𝜑 → ((𝑝 ∘𝑓 + 𝑞) ∈ (ℕ0 ↑𝑚 𝐼) ↔ (𝑝 ∘𝑓 + 𝑞):𝐼⟶ℕ0))
41	40	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → ((𝑝 ∘𝑓 + 𝑞) ∈ (ℕ0 ↑𝑚 𝐼) ↔ (𝑝 ∘𝑓 + 𝑞):𝐼⟶ℕ0))
42	39, 41	mpbird 167	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → (𝑝 ∘𝑓 + 𝑞) ∈ (ℕ0 ↑𝑚 𝐼))
43		simp-4l 543	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → 𝜑)
44		simplr 529	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → 𝑏 ∈ (ℕ0 ↑𝑚 𝐼))
45		eqid 2234	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
46	1, 2, 45, 3, 10, 12	psradd 14963	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑋 ∘𝑓 (+g‘𝑅)𝑌))
47	46	fveq1d 5677	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 + 𝑌)‘𝑏) = ((𝑋 ∘𝑓 (+g‘𝑅)𝑌)‘𝑏))
48	47	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → ((𝑋 + 𝑌)‘𝑏) = ((𝑋 ∘𝑓 (+g‘𝑅)𝑌)‘𝑏))
49		eqid 2234	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
50	1, 49, 14, 2, 10	psrelbasfi 14960	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋:(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
51	50	ffnd 5514	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 Fn (ℕ0 ↑𝑚 𝐼))
52	1, 49, 14, 2, 12	psrelbasfi 14960	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌:(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
53	52	ffnd 5514	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 Fn (ℕ0 ↑𝑚 𝐼))
54		fnmap 6902	. . . . . . . . . . . . 13 ⊢ ↑𝑚 Fn (V × V)
55	14	elexd 2829	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ V)
56		fnovex 6091	. . . . . . . . . . . . 13 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝐼 ∈ V) → (ℕ0 ↑𝑚 𝐼) ∈ V)
57	54, 28, 55, 56	mp3an12i 1378	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℕ0 ↑𝑚 𝐼) ∈ V)
58		inidm 3434	. . . . . . . . . . . 12 ⊢ ((ℕ0 ↑𝑚 𝐼) ∩ (ℕ0 ↑𝑚 𝐼)) = (ℕ0 ↑𝑚 𝐼)
59		eqidd 2235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑋‘𝑏) = (𝑋‘𝑏))
60		eqidd 2235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑌‘𝑏) = (𝑌‘𝑏))
61	4	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑅 ∈ Grp)
62	50	ffvelcdmda 5817	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑋‘𝑏) ∈ (Base‘𝑅))
63	52	ffvelcdmda 5817	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑌‘𝑏) ∈ (Base‘𝑅))
64	49, 45, 61, 62, 63	grpcld 13772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → ((𝑋‘𝑏)(+g‘𝑅)(𝑌‘𝑏)) ∈ (Base‘𝑅))
65	51, 53, 57, 57, 58, 59, 60, 64	ofvalg 6285	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → ((𝑋 ∘𝑓 (+g‘𝑅)𝑌)‘𝑏) = ((𝑋‘𝑏)(+g‘𝑅)(𝑌‘𝑏)))
66	48, 65	eqtrd 2267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → ((𝑋 + 𝑌)‘𝑏) = ((𝑋‘𝑏)(+g‘𝑅)(𝑌‘𝑏)))
67	43, 44, 66	syl2anc 411	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ((𝑋 + 𝑌)‘𝑏) = ((𝑋‘𝑏)(+g‘𝑅)(𝑌‘𝑏)))
68	32	ad3antrrr 492	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → 𝑝:𝐼⟶ℕ0)
69		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → 𝑒 ∈ 𝐼)
70	68, 69	ffvelcdmd 5818	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑝‘𝑒) ∈ ℕ0)
71	70	nn0red 9574	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑝‘𝑒) ∈ ℝ)
72	27	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → 𝑝 ∈ (ℕ0 ↑𝑚 𝐼))
73	30	biimpa 296	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑝:𝐼⟶ℕ0)
74	73	ffnd 5514	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑝 Fn 𝐼)
75	43, 72, 74	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → 𝑝 Fn 𝐼)
76	33	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → 𝑞 ∈ (ℕ0 ↑𝑚 𝐼))
77	34	biimpa 296	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑞:𝐼⟶ℕ0)
78	77	ffnd 5514	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑞 Fn 𝐼)
79	43, 76, 78	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → 𝑞 Fn 𝐼)
80	37	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → 𝐼 ∈ Fin)
81		eqidd 2235	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑝‘𝑒) = (𝑝‘𝑒))
82		eqidd 2235	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑞‘𝑒) = (𝑞‘𝑒))
83	36	ad3antrrr 492	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → 𝑞:𝐼⟶ℕ0)
84	83, 69	ffvelcdmd 5818	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑞‘𝑒) ∈ ℕ0)
85	70, 84	nn0addcld 9577	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → ((𝑝‘𝑒) + (𝑞‘𝑒)) ∈ ℕ0)
86	75, 79, 80, 80, 38, 81, 82, 85	ofvalg 6285	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → ((𝑝 ∘𝑓 + 𝑞)‘𝑒) = ((𝑝‘𝑒) + (𝑞‘𝑒)))
87	86, 85	eqeltrd 2311	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → ((𝑝 ∘𝑓 + 𝑞)‘𝑒) ∈ ℕ0)
88	87	nn0red 9574	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → ((𝑝 ∘𝑓 + 𝑞)‘𝑒) ∈ ℝ)
89		elmapi 6917	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (ℕ0 ↑𝑚 𝐼) → 𝑏:𝐼⟶ℕ0)
90	89	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑏:𝐼⟶ℕ0)
91	90	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → 𝑏:𝐼⟶ℕ0)
92	91, 69	ffvelcdmd 5818	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑏‘𝑒) ∈ ℕ0)
93	92	nn0red 9574	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑏‘𝑒) ∈ ℝ)
94		nn0addge1 9562	. . . . . . . . . . . . . . . 16 ⊢ (((𝑝‘𝑒) ∈ ℝ ∧ (𝑞‘𝑒) ∈ ℕ0) → (𝑝‘𝑒) ≤ ((𝑝‘𝑒) + (𝑞‘𝑒)))
95	71, 84, 94	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑝‘𝑒) ≤ ((𝑝‘𝑒) + (𝑞‘𝑒)))
96	95, 86	breqtrrd 4142	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑝‘𝑒) ≤ ((𝑝 ∘𝑓 + 𝑞)‘𝑒))
97		fveq2 5675	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑒 → ((𝑝 ∘𝑓 + 𝑞)‘𝑘) = ((𝑝 ∘𝑓 + 𝑞)‘𝑒))
98		fveq2 5675	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑒 → (𝑏‘𝑘) = (𝑏‘𝑒))
99	97, 98	breq12d 4127	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑒 → (((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘) ↔ ((𝑝 ∘𝑓 + 𝑞)‘𝑒) < (𝑏‘𝑒)))
100		simplr 529	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘))
101	99, 100, 69	rspcdva 2928	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → ((𝑝 ∘𝑓 + 𝑞)‘𝑒) < (𝑏‘𝑒))
102	71, 88, 93, 96, 101	lelttrd 8415	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑝‘𝑒) < (𝑏‘𝑒))
103	102	ralrimiva 2617	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ∀𝑒 ∈ 𝐼 (𝑝‘𝑒) < (𝑏‘𝑒))
104		fveq2 5675	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑒 → (𝑝‘𝑣) = (𝑝‘𝑒))
105		fveq2 5675	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑒 → (𝑏‘𝑣) = (𝑏‘𝑒))
106	104, 105	breq12d 4127	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑒 → ((𝑝‘𝑣) < (𝑏‘𝑣) ↔ (𝑝‘𝑒) < (𝑏‘𝑒)))
107	106	cbvralv 2780	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑏‘𝑣) ↔ ∀𝑒 ∈ 𝐼 (𝑝‘𝑒) < (𝑏‘𝑒))
108	103, 107	sylibr 134	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑏‘𝑣))
109		fveq1 5674	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑏 → (𝑢‘𝑣) = (𝑏‘𝑣))
110	109	breq2d 4126	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑏 → ((𝑝‘𝑣) < (𝑢‘𝑣) ↔ (𝑝‘𝑣) < (𝑏‘𝑣)))
111	110	ralbidv 2544	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑏 → (∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) ↔ ∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑏‘𝑣)))
112		fveqeq2 5684	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑏 → ((𝑋‘𝑢) = (0g‘𝑅) ↔ (𝑋‘𝑏) = (0g‘𝑅)))
113	111, 112	imbi12d 234	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑏 → ((∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)) ↔ (∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑏‘𝑣) → (𝑋‘𝑏) = (0g‘𝑅))))
114		simprr 533	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) → ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))
115	114	ad3antrrr 492	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))
116	113, 115, 44	rspcdva 2928	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → (∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑏‘𝑣) → (𝑋‘𝑏) = (0g‘𝑅)))
117	108, 116	mpd 13	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → (𝑋‘𝑏) = (0g‘𝑅))
118	117	oveq1d 6073	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ((𝑋‘𝑏)(+g‘𝑅)(𝑌‘𝑏)) = ((0g‘𝑅)(+g‘𝑅)(𝑌‘𝑏)))
119	67, 118	eqtrd 2267	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ((𝑋 + 𝑌)‘𝑏) = ((0g‘𝑅)(+g‘𝑅)(𝑌‘𝑏)))
120	43, 4	syl 14	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → 𝑅 ∈ Grp)
121	43, 44, 63	syl2anc 411	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → (𝑌‘𝑏) ∈ (Base‘𝑅))
122	49, 45, 15, 120, 121	grplidd 13791	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ((0g‘𝑅)(+g‘𝑅)(𝑌‘𝑏)) = (𝑌‘𝑏))
123	84	nn0red 9574	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑞‘𝑒) ∈ ℝ)
124		nn0addge2 9563	. . . . . . . . . . . . . 14 ⊢ (((𝑞‘𝑒) ∈ ℝ ∧ (𝑝‘𝑒) ∈ ℕ0) → (𝑞‘𝑒) ≤ ((𝑝‘𝑒) + (𝑞‘𝑒)))
125	123, 70, 124	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑞‘𝑒) ≤ ((𝑝‘𝑒) + (𝑞‘𝑒)))
126	125, 86	breqtrrd 4142	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑞‘𝑒) ≤ ((𝑝 ∘𝑓 + 𝑞)‘𝑒))
127	123, 88, 93, 126, 101	lelttrd 8415	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) ∧ 𝑒 ∈ 𝐼) → (𝑞‘𝑒) < (𝑏‘𝑒))
128	127	ralrimiva 2617	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ∀𝑒 ∈ 𝐼 (𝑞‘𝑒) < (𝑏‘𝑒))
129		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑒 → (𝑞‘𝑡) = (𝑞‘𝑒))
130		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑒 → (𝑏‘𝑡) = (𝑏‘𝑒))
131	129, 130	breq12d 4127	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑒 → ((𝑞‘𝑡) < (𝑏‘𝑡) ↔ (𝑞‘𝑒) < (𝑏‘𝑒)))
132	131	cbvralv 2780	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑏‘𝑡) ↔ ∀𝑒 ∈ 𝐼 (𝑞‘𝑒) < (𝑏‘𝑒))
133	128, 132	sylibr 134	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑏‘𝑡))
134		fveq1 5674	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑏 → (𝑠‘𝑡) = (𝑏‘𝑡))
135	134	breq2d 4126	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑏 → ((𝑞‘𝑡) < (𝑠‘𝑡) ↔ (𝑞‘𝑡) < (𝑏‘𝑡)))
136	135	ralbidv 2544	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑏 → (∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) ↔ ∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑏‘𝑡)))
137		fveqeq2 5684	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑏 → ((𝑌‘𝑠) = (0g‘𝑅) ↔ (𝑌‘𝑏) = (0g‘𝑅)))
138	136, 137	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑠 = 𝑏 → ((∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)) ↔ (∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑏‘𝑡) → (𝑌‘𝑏) = (0g‘𝑅))))
139		simprr 533	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))
140	139	ad2antrr 488	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))
141	138, 140, 44	rspcdva 2928	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → (∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑏‘𝑡) → (𝑌‘𝑏) = (0g‘𝑅)))
142	133, 141	mpd 13	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → (𝑌‘𝑏) = (0g‘𝑅))
143	119, 122, 142	3eqtrd 2271	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅))
144	143	ex 115	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) ∧ 𝑏 ∈ (ℕ0 ↑𝑚 𝐼)) → (∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅)))
145	144	ralrimiva 2617	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → ∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅)))
146		fveq1 5674	. . . . . . . 8 ⊢ (𝑎 = (𝑝 ∘𝑓 + 𝑞) → (𝑎‘𝑘) = ((𝑝 ∘𝑓 + 𝑞)‘𝑘))
147	146	breq1d 4124	. . . . . . 7 ⊢ (𝑎 = (𝑝 ∘𝑓 + 𝑞) → ((𝑎‘𝑘) < (𝑏‘𝑘) ↔ ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)))
148	147	ralbidv 2544	. . . . . 6 ⊢ (𝑎 = (𝑝 ∘𝑓 + 𝑞) → (∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) ↔ ∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘)))
149	148	rspceaimv 2932	. . . . 5 ⊢ (((𝑝 ∘𝑓 + 𝑞) ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 ((𝑝 ∘𝑓 + 𝑞)‘𝑘) < (𝑏‘𝑘) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅))) → ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅)))
150	42, 145, 149	syl2anc 411	. . . 4 ⊢ (((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) ∧ (𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑠 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑡 ∈ 𝐼 (𝑞‘𝑡) < (𝑠‘𝑡) → (𝑌‘𝑠) = (0g‘𝑅)))) → ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅)))
151	24, 150	rexlimddv 2667	. . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑢 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑣 ∈ 𝐼 (𝑝‘𝑣) < (𝑢‘𝑣) → (𝑋‘𝑢) = (0g‘𝑅)))) → ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅)))
152	19, 151	rexlimddv 2667	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅)))
153	6, 1, 2, 15, 7	mplelbascoe 14976	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ Grp) → ((𝑋 + 𝑌) ∈ 𝑈 ↔ ((𝑋 + 𝑌) ∈ (Base‘𝑆) ∧ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅)))))
154	14, 4, 153	syl2anc 411	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) ∈ 𝑈 ↔ ((𝑋 + 𝑌) ∈ (Base‘𝑆) ∧ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑏 ∈ (ℕ0 ↑𝑚 𝐼)(∀𝑘 ∈ 𝐼 (𝑎‘𝑘) < (𝑏‘𝑘) → ((𝑋 + 𝑌)‘𝑏) = (0g‘𝑅)))))
155	13, 152, 154	mpbir2and 953	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  Vcvv 2815   class class class wbr 4114   × cxp 4752   Fn wfn 5352  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058   ∘_𝑓 cof 6273   ↑_𝑚 cmap 6895  Fincfn 6988  ℝcr 8142   + caddc 8146   < clt 8324   ≤ cle 8325  ℕ₀cn0 9516  Basecbs 13299  +_gcplusg 13377  0_gc0g 13556  Grpcgrp 13758   mPwSer cmps 14938   mPoly cmpl 14939

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-apti 8258  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-rmo 2530  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-tp 3702  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-1st 6347  df-2nd 6348  df-1o 6660  df-er 6780  df-map 6897  df-ixp 6947  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-z 9598  df-uz 9875  df-fz 10365  df-struct 13301  df-ndx 13302  df-slot 13303  df-base 13305  df-sets 13306  df-iress 13307  df-plusg 13390  df-mulr 13391  df-sca 13393  df-vsca 13394  df-tset 13396  df-rest 13541  df-topn 13542  df-0g 13558  df-topgen 13560  df-pt 13561  df-mgm 13622  df-sgrp 13668  df-mnd 13681  df-grp 13761  df-psr 14940  df-mplcoe 14941

This theorem is referenced by:  mplsubgfi  14985