Theorem mplsubgfilemcl 15232

Description: Lemma for mplsubgfi 15234. 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 𝑐 𝑝 𝑞 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		mplsubgfilemcl.p	. . 3 ⊢ + = (+g‘𝑆)
4		mplsubg.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
5	4	grpmgmd 13889	. . 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 15229	. . . 4 ⊢ 𝑈 ⊆ (Base‘𝑆)
9		mplsubgfilemcl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
10	8, 9	sselid 3246	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑆))
11		mplsubgfilemcl.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
12	8, 11	sselid 3246	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑆))
13	1, 2, 3, 5, 10, 12	psraddcl 15193	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (Base‘𝑆))
14		eqid 2238	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
15	6, 1, 2, 14, 7	mplelbas 15224	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ (Base‘𝑆) ∧ 𝑋 finSupp (0g‘𝑅)))
16	9, 15	sylib 122	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝑆) ∧ 𝑋 finSupp (0g‘𝑅)))
17	16	simprd 114	. . . 4 ⊢ (𝜑 → 𝑋 finSupp (0g‘𝑅))
18	17	fsuppfiss 7300	. . 3 ⊢ (𝜑 → ∃𝑎(𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)}))
19	6, 1, 2, 14, 7	mplelbas 15224	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑈 ↔ (𝑌 ∈ (Base‘𝑆) ∧ 𝑌 finSupp (0g‘𝑅)))
20	11, 19	sylib 122	. . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ (Base‘𝑆) ∧ 𝑌 finSupp (0g‘𝑅)))
21	20	simprd 114	. . . . . 6 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅))
22	21	fsuppfiss 7300	. . . . 5 ⊢ (𝜑 → ∃𝑏(𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)}))
23	22	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) → ∃𝑏(𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)}))
24		simplr2 1071	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → 𝑎 ⊆ dom 𝑋)
25		eqid 2238	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
26		mplsubg.i	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ Fin)
27	1, 25, 26, 2, 10	psrelbasfi 15189	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
28	27	fdmd 5543	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑋 = (ℕ0 ↑𝑚 𝐼))
29	28	sseq2d 3278	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ⊆ dom 𝑋 ↔ 𝑎 ⊆ (ℕ0 ↑𝑚 𝐼)))
30	29	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → (𝑎 ⊆ dom 𝑋 ↔ 𝑎 ⊆ (ℕ0 ↑𝑚 𝐼)))
31	24, 30	mpbid 147	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → 𝑎 ⊆ (ℕ0 ↑𝑚 𝐼))
32		simplr1 1070	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → 𝑎 ∈ Fin)
33		simpr2 1035	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → 𝑏 ⊆ dom 𝑌)
34	1, 25, 26, 2, 12	psrelbasfi 15189	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
35	34	fdmd 5543	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑌 = (ℕ0 ↑𝑚 𝐼))
36	35	sseq2d 3278	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ⊆ dom 𝑌 ↔ 𝑏 ⊆ (ℕ0 ↑𝑚 𝐼)))
37	36	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → (𝑏 ⊆ dom 𝑌 ↔ 𝑏 ⊆ (ℕ0 ↑𝑚 𝐼)))
38	33, 37	mpbid 147	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → 𝑏 ⊆ (ℕ0 ↑𝑚 𝐼))
39		simpr1 1034	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → 𝑏 ∈ Fin)
40	26	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → 𝐼 ∈ Fin)
41	31, 32, 38, 39, 40	psrbagmax 15183	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → ∃𝑐((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐)))
42	13	ad3antrrr 496	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑋 + 𝑌) ∈ (Base‘𝑆))
43	25, 14	grpidcl 13892	. . . . . . . 8 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ (Base‘𝑅))
44	4, 43	syl 14	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
45	44	ad3antrrr 496	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (0g‘𝑅) ∈ (Base‘𝑅))
46	1, 25, 26, 2, 13	psrelbasfi 15189	. . . . . . . 8 ⊢ (𝜑 → (𝑋 + 𝑌):(ℕ0 ↑𝑚 𝐼)⟶(Base‘𝑅))
47	46	ffund 5540	. . . . . . 7 ⊢ (𝜑 → Fun (𝑋 + 𝑌))
48	47	ad3antrrr 496	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → Fun (𝑋 + 𝑌))
49		simprlr 544	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → 𝑐 ∈ Fin)
50		simprll 543	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → 𝑐 ⊆ (ℕ0 ↑𝑚 𝐼))
51	46	fdmd 5543	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑋 + 𝑌) = (ℕ0 ↑𝑚 𝐼))
52	51	sseq2d 3278	. . . . . . . 8 ⊢ (𝜑 → (𝑐 ⊆ dom (𝑋 + 𝑌) ↔ 𝑐 ⊆ (ℕ0 ↑𝑚 𝐼)))
53	52	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑐 ⊆ dom (𝑋 + 𝑌) ↔ 𝑐 ⊆ (ℕ0 ↑𝑚 𝐼)))
54	50, 53	mpbird 167	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → 𝑐 ⊆ dom (𝑋 + 𝑌))
55		fvelima 5757	. . . . . . . . . 10 ⊢ ((Fun (𝑋 + 𝑌) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) → ∃𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐)((𝑋 + 𝑌)‘𝑞) = 𝑝)
56	48, 55	sylan 283	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) → ∃𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐)((𝑋 + 𝑌)‘𝑞) = 𝑝)
57		eqid 2238	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝑅) = (+g‘𝑅)
58	1, 2, 57, 3, 10, 12	psradd 15192	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑋 ∘𝑓 (+g‘𝑅)𝑌))
59	58	fveq1d 5700	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑋 + 𝑌)‘𝑞) = ((𝑋 ∘𝑓 (+g‘𝑅)𝑌)‘𝑞))
60	59	ad5antr 500	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → ((𝑋 + 𝑌)‘𝑞) = ((𝑋 ∘𝑓 (+g‘𝑅)𝑌)‘𝑞))
61		simp-5l 549	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → 𝜑)
62		simprl 535	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → 𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐))
63	62	eldifad 3231	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → 𝑞 ∈ dom (𝑋 + 𝑌))
64	51	eleq2d 2308	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑞 ∈ dom (𝑋 + 𝑌) ↔ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)))
65	64	ad5antr 500	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑞 ∈ dom (𝑋 + 𝑌) ↔ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)))
66	63, 65	mpbid 147	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → 𝑞 ∈ (ℕ0 ↑𝑚 𝐼))
67	27	ffnd 5537	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 Fn (ℕ0 ↑𝑚 𝐼))
68	34	ffnd 5537	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 Fn (ℕ0 ↑𝑚 𝐼))
69	9, 67	fndmexd 5584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℕ0 ↑𝑚 𝐼) ∈ V)
70		inidm 3440	. . . . . . . . . . . . . 14 ⊢ ((ℕ0 ↑𝑚 𝐼) ∩ (ℕ0 ↑𝑚 𝐼)) = (ℕ0 ↑𝑚 𝐼)
71		eqidd 2239	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑋‘𝑞) = (𝑋‘𝑞))
72		eqidd 2239	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑌‘𝑞) = (𝑌‘𝑞))
73	4	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑅 ∈ Grp)
74	27	ffvelcdmda 5846	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑋‘𝑞) ∈ (Base‘𝑅))
75	34	ffvelcdmda 5846	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑌‘𝑞) ∈ (Base‘𝑅))
76	25, 57, 73, 74, 75	grpcld 13877	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)) → ((𝑋‘𝑞)(+g‘𝑅)(𝑌‘𝑞)) ∈ (Base‘𝑅))
77	67, 68, 69, 69, 70, 71, 72, 76	ofvalg 6316	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ (ℕ0 ↑𝑚 𝐼)) → ((𝑋 ∘𝑓 (+g‘𝑅)𝑌)‘𝑞) = ((𝑋‘𝑞)(+g‘𝑅)(𝑌‘𝑞)))
78	61, 66, 77	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → ((𝑋 ∘𝑓 (+g‘𝑅)𝑌)‘𝑞) = ((𝑋‘𝑞)(+g‘𝑅)(𝑌‘𝑞)))
79	60, 78	eqtrd 2271	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → ((𝑋 + 𝑌)‘𝑞) = ((𝑋‘𝑞)(+g‘𝑅)(𝑌‘𝑞)))
80		simprr 537	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → ((𝑋 + 𝑌)‘𝑞) = 𝑝)
81		simprrl 545	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → 𝑎 ⊆ 𝑐)
82	81	sscond 3366	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐) ⊆ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑎))
83		imass2 5166	. . . . . . . . . . . . . . . . 17 ⊢ (((ℕ0 ↑𝑚 𝐼) ∖ 𝑐) ⊆ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑎) → (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)) ⊆ (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑎)))
84	82, 83	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)) ⊆ (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑎)))
85		simpr3 1036	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) → (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})
86	85	ad2antrr 492	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})
87	28	difeq1d 3346	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (dom 𝑋 ∖ 𝑎) = ((ℕ0 ↑𝑚 𝐼) ∖ 𝑎))
88	87	imaeq2d 5129	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑋 “ (dom 𝑋 ∖ 𝑎)) = (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑎)))
89	88	sseq1d 3277	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)} ↔ (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑎)) ⊆ {(0g‘𝑅)}))
90	89	ad3antrrr 496	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → ((𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)} ↔ (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑎)) ⊆ {(0g‘𝑅)}))
91	86, 90	mpbid 147	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑎)) ⊆ {(0g‘𝑅)})
92	84, 91	sstrd 3258	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)) ⊆ {(0g‘𝑅)})
93	92	ad2antrr 492	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)) ⊆ {(0g‘𝑅)})
94	61, 67	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → 𝑋 Fn (ℕ0 ↑𝑚 𝐼))
95	51	difeq1d 3346	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (dom (𝑋 + 𝑌) ∖ 𝑐) = ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐))
96	95	eleq2d 2308	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ↔ 𝑞 ∈ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)))
97	96	ad5antr 500	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ↔ 𝑞 ∈ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)))
98	62, 97	mpbid 147	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → 𝑞 ∈ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐))
99	94, 66, 98	fnfvimad 5957	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑋‘𝑞) ∈ (𝑋 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)))
100	93, 99	sseldd 3249	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑋‘𝑞) ∈ {(0g‘𝑅)})
101		elsni 3727	. . . . . . . . . . . . 13 ⊢ ((𝑋‘𝑞) ∈ {(0g‘𝑅)} → (𝑋‘𝑞) = (0g‘𝑅))
102	100, 101	syl 14	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑋‘𝑞) = (0g‘𝑅))
103		simprrr 546	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → 𝑏 ⊆ 𝑐)
104	103	sscond 3366	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐) ⊆ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑏))
105		imass2 5166	. . . . . . . . . . . . . . . . 17 ⊢ (((ℕ0 ↑𝑚 𝐼) ∖ 𝑐) ⊆ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑏) → (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)) ⊆ (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑏)))
106	104, 105	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)) ⊆ (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑏)))
107		simplr3 1072	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})
108	35	difeq1d 3346	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (dom 𝑌 ∖ 𝑏) = ((ℕ0 ↑𝑚 𝐼) ∖ 𝑏))
109	108	imaeq2d 5129	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑌 “ (dom 𝑌 ∖ 𝑏)) = (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑏)))
110	109	sseq1d 3277	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)} ↔ (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑏)) ⊆ {(0g‘𝑅)}))
111	110	ad3antrrr 496	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → ((𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)} ↔ (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑏)) ⊆ {(0g‘𝑅)}))
112	107, 111	mpbid 147	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑏)) ⊆ {(0g‘𝑅)})
113	106, 112	sstrd 3258	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)) ⊆ {(0g‘𝑅)})
114	113	ad2antrr 492	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)) ⊆ {(0g‘𝑅)})
115	61, 68	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → 𝑌 Fn (ℕ0 ↑𝑚 𝐼))
116	115, 66, 98	fnfvimad 5957	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑌‘𝑞) ∈ (𝑌 “ ((ℕ0 ↑𝑚 𝐼) ∖ 𝑐)))
117	114, 116	sseldd 3249	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑌‘𝑞) ∈ {(0g‘𝑅)})
118		elsni 3727	. . . . . . . . . . . . 13 ⊢ ((𝑌‘𝑞) ∈ {(0g‘𝑅)} → (𝑌‘𝑞) = (0g‘𝑅))
119	117, 118	syl 14	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → (𝑌‘𝑞) = (0g‘𝑅))
120	102, 119	oveq12d 6106	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → ((𝑋‘𝑞)(+g‘𝑅)(𝑌‘𝑞)) = ((0g‘𝑅)(+g‘𝑅)(0g‘𝑅)))
121	79, 80, 120	3eqtr3d 2279	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → 𝑝 = ((0g‘𝑅)(+g‘𝑅)(0g‘𝑅)))
122	25, 57, 14, 4, 44	grpridd 13897	. . . . . . . . . . . 12 ⊢ (𝜑 → ((0g‘𝑅)(+g‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
123		elsn2g 3742	. . . . . . . . . . . . 13 ⊢ ((0g‘𝑅) ∈ (Base‘𝑅) → (((0g‘𝑅)(+g‘𝑅)(0g‘𝑅)) ∈ {(0g‘𝑅)} ↔ ((0g‘𝑅)(+g‘𝑅)(0g‘𝑅)) = (0g‘𝑅)))
124	44, 123	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (((0g‘𝑅)(+g‘𝑅)(0g‘𝑅)) ∈ {(0g‘𝑅)} ↔ ((0g‘𝑅)(+g‘𝑅)(0g‘𝑅)) = (0g‘𝑅)))
125	122, 124	mpbird 167	. . . . . . . . . . 11 ⊢ (𝜑 → ((0g‘𝑅)(+g‘𝑅)(0g‘𝑅)) ∈ {(0g‘𝑅)})
126	61, 125	syl 14	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → ((0g‘𝑅)(+g‘𝑅)(0g‘𝑅)) ∈ {(0g‘𝑅)})
127	121, 126	eqeltrd 2315	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) ∧ (𝑞 ∈ (dom (𝑋 + 𝑌) ∖ 𝑐) ∧ ((𝑋 + 𝑌)‘𝑞) = 𝑝)) → 𝑝 ∈ {(0g‘𝑅)})
128	56, 127	rexlimddv 2673	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) ∧ 𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐))) → 𝑝 ∈ {(0g‘𝑅)})
129	128	ex 115	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑝 ∈ ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐)) → 𝑝 ∈ {(0g‘𝑅)}))
130	129	ssrdv 3254	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → ((𝑋 + 𝑌) “ (dom (𝑋 + 𝑌) ∖ 𝑐)) ⊆ {(0g‘𝑅)})
131	42, 45, 48, 49, 54, 130	isfsuppfid 7297	. . . . 5 ⊢ ((((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) ∧ ((𝑐 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑐 ∈ Fin) ∧ (𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐))) → (𝑋 + 𝑌) finSupp (0g‘𝑅))
132	41, 131	exlimddv 1954	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) ∧ (𝑏 ∈ Fin ∧ 𝑏 ⊆ dom 𝑌 ∧ (𝑌 “ (dom 𝑌 ∖ 𝑏)) ⊆ {(0g‘𝑅)})) → (𝑋 + 𝑌) finSupp (0g‘𝑅))
133	23, 132	exlimddv 1954	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ Fin ∧ 𝑎 ⊆ dom 𝑋 ∧ (𝑋 “ (dom 𝑋 ∖ 𝑎)) ⊆ {(0g‘𝑅)})) → (𝑋 + 𝑌) finSupp (0g‘𝑅))
134	18, 133	exlimddv 1954	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) finSupp (0g‘𝑅))
135	6, 1, 2, 14, 7	mplelbas 15224	. 2 ⊢ ((𝑋 + 𝑌) ∈ 𝑈 ↔ ((𝑋 + 𝑌) ∈ (Base‘𝑆) ∧ (𝑋 + 𝑌) finSupp (0g‘𝑅)))
136	13, 134, 135	sylanbrc 421	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821   ∖ cdif 3217   ⊆ wss 3220  {csn 3709   class class class wbr 4131  dom cdm 4776   “ cima 4779  Fun wfun 5374   Fn wfn 5375  ‘cfv 5380  (class class class)co 6088   ∘_𝑓 cof 6303   ↑_𝑚 cmap 6927  Fincfn 7028   finSupp cfsupp 7292  ℕ₀cn0 9574  Basecbs 13409  +_gcplusg 13489  0_gc0g 13668  Grpcgrp 13863   mPwSer cmps 15153   mPoly cmpl 15155

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 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof 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-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-frec 6667  df-1o 6692  df-er 6812  df-map 6929  df-ixp 6987  df-en 7029  df-fin 7031  df-fsuppfi 7293  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9654  df-uz 9937  df-rp 10071  df-fz 10428  df-seqfrec 10905  df-exp 10996  df-cj 11628  df-re 11629  df-im 11630  df-rsqrt 11785  df-abs 11786  df-struct 13411  df-ndx 13412  df-slot 13413  df-base 13415  df-sets 13416  df-iress 13417  df-plusg 13502  df-mulr 13503  df-sca 13505  df-vsca 13506  df-tset 13508  df-rest 13653  df-topn 13654  df-0g 13670  df-topgen 13672  df-pt 13673  df-mgm 13734  df-sgrp 13775  df-mnd 13788  df-grp 13866  df-psr 15156  df-mpl 15158

This theorem is used by:  mplsubgfi  15234