Theorem gsumwun 33176

Description: In a commutative ring, a group sum of a word 𝑊 of characters taken from two submonoids 𝐸 and 𝐹 can be written as a simple sum. (Contributed by Thierry Arnoux, 6-Oct-2025.)

Hypotheses

Ref	Expression
gsumwun.p	⊢ + = (+g‘𝑀)
gsumwun.m	⊢ (𝜑 → 𝑀 ∈ CMnd)
gsumwun.e	⊢ (𝜑 → 𝐸 ∈ (SubMnd‘𝑀))
gsumwun.f	⊢ (𝜑 → 𝐹 ∈ (SubMnd‘𝑀))
gsumwun.w	⊢ (𝜑 → 𝑊 ∈ Word (𝐸 ∪ 𝐹))

Assertion

Ref	Expression
gsumwun	⊢ (𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑊) = (𝑒 + 𝑓))

Distinct variable groups:   + ,𝑒,𝑓   𝑒,𝐸,𝑓   𝑒,𝐹,𝑓   𝑒,𝑀,𝑓   𝑒,𝑊,𝑓   𝜑,𝑒,𝑓

Proof of Theorem gsumwun

Dummy variables 𝑖 𝑗 𝑣 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumwun.w	. 2 ⊢ (𝜑 → 𝑊 ∈ Word (𝐸 ∪ 𝐹))
2		oveq2 7378	. . . . . 6 ⊢ (𝑣 = ∅ → (𝑀 Σg 𝑣) = (𝑀 Σg ∅))
3	2	eqeq1d 2739	. . . . 5 ⊢ (𝑣 = ∅ → ((𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ (𝑀 Σg ∅) = (𝑒 + 𝑓)))
4	3	2rexbidv 3203	. . . 4 ⊢ (𝑣 = ∅ → (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg ∅) = (𝑒 + 𝑓)))
5	4	imbi2d 340	. . 3 ⊢ (𝑣 = ∅ → ((𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓)) ↔ (𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg ∅) = (𝑒 + 𝑓))))
6		oveq2 7378	. . . . . 6 ⊢ (𝑣 = 𝑤 → (𝑀 Σg 𝑣) = (𝑀 Σg 𝑤))
7	6	eqeq1d 2739	. . . . 5 ⊢ (𝑣 = 𝑤 → ((𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)))
8	7	2rexbidv 3203	. . . 4 ⊢ (𝑣 = 𝑤 → (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑤) = (𝑒 + 𝑓)))
9	8	imbi2d 340	. . 3 ⊢ (𝑣 = 𝑤 → ((𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓)) ↔ (𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑤) = (𝑒 + 𝑓))))
10		oveq1 7377	. . . . . . 7 ⊢ (𝑒 = 𝑖 → (𝑒 + 𝑓) = (𝑖 + 𝑓))
11	10	eqeq2d 2748	. . . . . 6 ⊢ (𝑒 = 𝑖 → ((𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ (𝑀 Σg 𝑣) = (𝑖 + 𝑓)))
12		oveq2 7378	. . . . . . 7 ⊢ (𝑓 = 𝑗 → (𝑖 + 𝑓) = (𝑖 + 𝑗))
13	12	eqeq2d 2748	. . . . . 6 ⊢ (𝑓 = 𝑗 → ((𝑀 Σg 𝑣) = (𝑖 + 𝑓) ↔ (𝑀 Σg 𝑣) = (𝑖 + 𝑗)))
14	11, 13	cbvrex2vw 3221	. . . . 5 ⊢ (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑖 + 𝑗))
15		oveq2 7378	. . . . . . 7 ⊢ (𝑣 = (𝑤 ++ ⟨“𝑥”⟩) → (𝑀 Σg 𝑣) = (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)))
16	15	eqeq1d 2739	. . . . . 6 ⊢ (𝑣 = (𝑤 ++ ⟨“𝑥”⟩) → ((𝑀 Σg 𝑣) = (𝑖 + 𝑗) ↔ (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗)))
17	16	2rexbidv 3203	. . . . 5 ⊢ (𝑣 = (𝑤 ++ ⟨“𝑥”⟩) → (∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑖 + 𝑗) ↔ ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗)))
18	14, 17	bitrid 283	. . . 4 ⊢ (𝑣 = (𝑤 ++ ⟨“𝑥”⟩) → (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗)))
19	18	imbi2d 340	. . 3 ⊢ (𝑣 = (𝑤 ++ ⟨“𝑥”⟩) → ((𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓)) ↔ (𝜑 → ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗))))
20		oveq2 7378	. . . . . 6 ⊢ (𝑣 = 𝑊 → (𝑀 Σg 𝑣) = (𝑀 Σg 𝑊))
21	20	eqeq1d 2739	. . . . 5 ⊢ (𝑣 = 𝑊 → ((𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ (𝑀 Σg 𝑊) = (𝑒 + 𝑓)))
22	21	2rexbidv 3203	. . . 4 ⊢ (𝑣 = 𝑊 → (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑊) = (𝑒 + 𝑓)))
23	22	imbi2d 340	. . 3 ⊢ (𝑣 = 𝑊 → ((𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓)) ↔ (𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑊) = (𝑒 + 𝑓))))
24		oveq1 7377	. . . . 5 ⊢ (𝑒 = (0g‘𝑀) → (𝑒 + 𝑓) = ((0g‘𝑀) + 𝑓))
25	24	eqeq2d 2748	. . . 4 ⊢ (𝑒 = (0g‘𝑀) → ((𝑀 Σg ∅) = (𝑒 + 𝑓) ↔ (𝑀 Σg ∅) = ((0g‘𝑀) + 𝑓)))
26		oveq2 7378	. . . . 5 ⊢ (𝑓 = (0g‘𝑀) → ((0g‘𝑀) + 𝑓) = ((0g‘𝑀) + (0g‘𝑀)))
27	26	eqeq2d 2748	. . . 4 ⊢ (𝑓 = (0g‘𝑀) → ((𝑀 Σg ∅) = ((0g‘𝑀) + 𝑓) ↔ (𝑀 Σg ∅) = ((0g‘𝑀) + (0g‘𝑀))))
28		gsumwun.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (SubMnd‘𝑀))
29		eqid 2737	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
30	29	subm0cl 18750	. . . . 5 ⊢ (𝐸 ∈ (SubMnd‘𝑀) → (0g‘𝑀) ∈ 𝐸)
31	28, 30	syl 17	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝐸)
32		gsumwun.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubMnd‘𝑀))
33	29	subm0cl 18750	. . . . 5 ⊢ (𝐹 ∈ (SubMnd‘𝑀) → (0g‘𝑀) ∈ 𝐹)
34	32, 33	syl 17	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝐹)
35	29	gsum0 18623	. . . . 5 ⊢ (𝑀 Σg ∅) = (0g‘𝑀)
36		gsumwun.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ CMnd)
37	36	cmnmndd 19750	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Mnd)
38		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
39	38, 29	mndidcl 18688	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
40		gsumwun.p	. . . . . . 7 ⊢ + = (+g‘𝑀)
41	38, 40, 29	mndlid 18693	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ (0g‘𝑀) ∈ (Base‘𝑀)) → ((0g‘𝑀) + (0g‘𝑀)) = (0g‘𝑀))
42	37, 39, 41	syl2anc2 586	. . . . 5 ⊢ (𝜑 → ((0g‘𝑀) + (0g‘𝑀)) = (0g‘𝑀))
43	35, 42	eqtr4id 2791	. . . 4 ⊢ (𝜑 → (𝑀 Σg ∅) = ((0g‘𝑀) + (0g‘𝑀)))
44	25, 27, 31, 34, 43	2rspcedvdw 3592	. . 3 ⊢ (𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg ∅) = (𝑒 + 𝑓))
45		oveq1 7377	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑒 + 𝑥) → (𝑖 + 𝑗) = ((𝑒 + 𝑥) + 𝑗))
46	45	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑖 = (𝑒 + 𝑥) → ((𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗) ↔ (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = ((𝑒 + 𝑥) + 𝑗)))
47		oveq2 7378	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑓 → ((𝑒 + 𝑥) + 𝑗) = ((𝑒 + 𝑥) + 𝑓))
48	47	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑗 = 𝑓 → ((𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = ((𝑒 + 𝑥) + 𝑗) ↔ (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = ((𝑒 + 𝑥) + 𝑓)))
49	28	ad6antr 737	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐸) → 𝐸 ∈ (SubMnd‘𝑀))
50		simp-4r 784	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐸) → 𝑒 ∈ 𝐸)
51		simpr 484	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐸) → 𝑥 ∈ 𝐸)
52	40, 49, 50, 51	submcld 33134	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐸) → (𝑒 + 𝑥) ∈ 𝐸)
53		simpllr 776	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐸) → 𝑓 ∈ 𝐹)
54	37	ad5antr 735	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → 𝑀 ∈ Mnd)
55	38	submss 18748	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∈ (SubMnd‘𝑀) → 𝐸 ⊆ (Base‘𝑀))
56	28, 55	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ⊆ (Base‘𝑀))
57	38	submss 18748	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (SubMnd‘𝑀) → 𝐹 ⊆ (Base‘𝑀))
58	32, 57	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝑀))
59	56, 58	unssd 4146	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ⊆ (Base‘𝑀))
60		sswrd 14459	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 ∪ 𝐹) ⊆ (Base‘𝑀) → Word (𝐸 ∪ 𝐹) ⊆ Word (Base‘𝑀))
61	59, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Word (𝐸 ∪ 𝐹) ⊆ Word (Base‘𝑀))
62	61	sselda 3935	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → 𝑤 ∈ Word (Base‘𝑀))
63	62	ad4antr 733	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → 𝑤 ∈ Word (Base‘𝑀))
64	59	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → (𝐸 ∪ 𝐹) ⊆ (Base‘𝑀))
65	64	sselda 3935	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) → 𝑥 ∈ (Base‘𝑀))
66	65	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → 𝑥 ∈ (Base‘𝑀))
67	38, 40	gsumccatsn 18782	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ Mnd ∧ 𝑤 ∈ Word (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = ((𝑀 Σg 𝑤) + 𝑥))
68	54, 63, 66, 67	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = ((𝑀 Σg 𝑤) + 𝑥))
69		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → (𝑀 Σg 𝑤) = (𝑒 + 𝑓))
70	69	oveq1d 7385	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → ((𝑀 Σg 𝑤) + 𝑥) = ((𝑒 + 𝑓) + 𝑥))
71	56	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) → 𝐸 ⊆ (Base‘𝑀))
72	71	sselda 3935	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ (Base‘𝑀))
73	72	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → 𝑒 ∈ (Base‘𝑀))
74	58	ad3antrrr 731	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) → 𝐹 ⊆ (Base‘𝑀))
75	74	sselda 3935	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) → 𝑓 ∈ (Base‘𝑀))
76	75	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → 𝑓 ∈ (Base‘𝑀))
77	36	ad5antr 735	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → 𝑀 ∈ CMnd)
78	38, 40	cmncom 19744	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ CMnd ∧ 𝑓 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑓 + 𝑥) = (𝑥 + 𝑓))
79	77, 76, 66, 78	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → (𝑓 + 𝑥) = (𝑥 + 𝑓))
80	38, 40, 54, 73, 76, 66, 79	mnd32g 18685	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → ((𝑒 + 𝑓) + 𝑥) = ((𝑒 + 𝑥) + 𝑓))
81	68, 70, 80	3eqtrd 2776	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = ((𝑒 + 𝑥) + 𝑓))
82	81	adantr 480	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐸) → (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = ((𝑒 + 𝑥) + 𝑓))
83	46, 48, 52, 53, 82	2rspcedvdw 3592	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐸) → ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗))
84		oveq1 7377	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑒 → (𝑖 + 𝑗) = (𝑒 + 𝑗))
85	84	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑖 = 𝑒 → ((𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗) ↔ (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑒 + 𝑗)))
86		oveq2 7378	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑓 + 𝑥) → (𝑒 + 𝑗) = (𝑒 + (𝑓 + 𝑥)))
87	86	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑗 = (𝑓 + 𝑥) → ((𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑒 + 𝑗) ↔ (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑒 + (𝑓 + 𝑥))))
88		simp-4r 784	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐹) → 𝑒 ∈ 𝐸)
89	32	ad6antr 737	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐹) → 𝐹 ∈ (SubMnd‘𝑀))
90		simpllr 776	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐹) → 𝑓 ∈ 𝐹)
91		simpr 484	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹)
92	40, 89, 90, 91	submcld 33134	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐹) → (𝑓 + 𝑥) ∈ 𝐹)
93	38, 40, 54, 73, 76, 66	mndassd 33122	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → ((𝑒 + 𝑓) + 𝑥) = (𝑒 + (𝑓 + 𝑥)))
94	68, 70, 93	3eqtrd 2776	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑒 + (𝑓 + 𝑥)))
95	94	adantr 480	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐹) → (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑒 + (𝑓 + 𝑥)))
96	85, 87, 88, 92, 95	2rspcedvdw 3592	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) ∧ 𝑥 ∈ 𝐹) → ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗))
97		elun 4107	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐸 ∪ 𝐹) ↔ (𝑥 ∈ 𝐸 ∨ 𝑥 ∈ 𝐹))
98	97	biimpi 216	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐸 ∪ 𝐹) → (𝑥 ∈ 𝐸 ∨ 𝑥 ∈ 𝐹))
99	98	ad4antlr 734	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → (𝑥 ∈ 𝐸 ∨ 𝑥 ∈ 𝐹))
100	83, 96, 99	mpjaodan 961	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗))
101	100	r19.29ffa 32563	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗))
102	101	ex 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) → (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑤) = (𝑒 + 𝑓) → ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗)))
103	102	expl 457	. . . . 5 ⊢ (𝜑 → ((𝑤 ∈ Word (𝐸 ∪ 𝐹) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) → (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑤) = (𝑒 + 𝑓) → ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗))))
104	103	com12 32	. . . 4 ⊢ ((𝑤 ∈ Word (𝐸 ∪ 𝐹) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) → (𝜑 → (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑤) = (𝑒 + 𝑓) → ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗))))
105	104	a2d 29	. . 3 ⊢ ((𝑤 ∈ Word (𝐸 ∪ 𝐹) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) → ((𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → (𝜑 → ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗))))
106	5, 9, 19, 23, 44, 105	wrdind 14659	. 2 ⊢ (𝑊 ∈ Word (𝐸 ∪ 𝐹) → (𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑊) = (𝑒 + 𝑓)))
107	1, 106	mpcom 38	1 ⊢ (𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑊) = (𝑒 + 𝑓))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  ‘cfv 6502  (class class class)co 7370  Word cword 14450   ++ cconcat 14507  ⟨“cs1 14533  Basecbs 17150  +_gcplusg 17191  0_gc0g 17373   Σ_g cgsu 17374  Mndcmnd 18673  SubMndcsubmnd 18721  CMndccmn 19726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-n0 12416  df-xnn0 12489  df-z 12503  df-uz 12766  df-fz 13438  df-fzo 13585  df-seq 13939  df-hash 14268  df-word 14451  df-lsw 14500  df-concat 14508  df-s1 14534  df-substr 14579  df-pfx 14609  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-0g 17375  df-gsum 17376  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-submnd 18723  df-cmn 19728

This theorem is referenced by:  elrgspnsubrunlem2  33348