Theorem gsumwun 33137

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 7366	. . . . . 6 ⊢ (𝑣 = ∅ → (𝑀 Σg 𝑣) = (𝑀 Σg ∅))
3	2	eqeq1d 2737	. . . . 5 ⊢ (𝑣 = ∅ → ((𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ (𝑀 Σg ∅) = (𝑒 + 𝑓)))
4	3	2rexbidv 3200	. . . 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 7366	. . . . . 6 ⊢ (𝑣 = 𝑤 → (𝑀 Σg 𝑣) = (𝑀 Σg 𝑤))
7	6	eqeq1d 2737	. . . . 5 ⊢ (𝑣 = 𝑤 → ((𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)))
8	7	2rexbidv 3200	. . . 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 7365	. . . . . . 7 ⊢ (𝑒 = 𝑖 → (𝑒 + 𝑓) = (𝑖 + 𝑓))
11	10	eqeq2d 2746	. . . . . 6 ⊢ (𝑒 = 𝑖 → ((𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ (𝑀 Σg 𝑣) = (𝑖 + 𝑓)))
12		oveq2 7366	. . . . . . 7 ⊢ (𝑓 = 𝑗 → (𝑖 + 𝑓) = (𝑖 + 𝑗))
13	12	eqeq2d 2746	. . . . . 6 ⊢ (𝑓 = 𝑗 → ((𝑀 Σg 𝑣) = (𝑖 + 𝑓) ↔ (𝑀 Σg 𝑣) = (𝑖 + 𝑗)))
14	11, 13	cbvrex2vw 3218	. . . . 5 ⊢ (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑖 + 𝑗))
15		oveq2 7366	. . . . . . 7 ⊢ (𝑣 = (𝑤 ++ ⟨“𝑥”⟩) → (𝑀 Σg 𝑣) = (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)))
16	15	eqeq1d 2737	. . . . . 6 ⊢ (𝑣 = (𝑤 ++ ⟨“𝑥”⟩) → ((𝑀 Σg 𝑣) = (𝑖 + 𝑗) ↔ (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗)))
17	16	2rexbidv 3200	. . . . 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 7366	. . . . . 6 ⊢ (𝑣 = 𝑊 → (𝑀 Σg 𝑣) = (𝑀 Σg 𝑊))
21	20	eqeq1d 2737	. . . . 5 ⊢ (𝑣 = 𝑊 → ((𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ (𝑀 Σg 𝑊) = (𝑒 + 𝑓)))
22	21	2rexbidv 3200	. . . 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 7365	. . . . 5 ⊢ (𝑒 = (0g‘𝑀) → (𝑒 + 𝑓) = ((0g‘𝑀) + 𝑓))
25	24	eqeq2d 2746	. . . 4 ⊢ (𝑒 = (0g‘𝑀) → ((𝑀 Σg ∅) = (𝑒 + 𝑓) ↔ (𝑀 Σg ∅) = ((0g‘𝑀) + 𝑓)))
26		oveq2 7366	. . . . 5 ⊢ (𝑓 = (0g‘𝑀) → ((0g‘𝑀) + 𝑓) = ((0g‘𝑀) + (0g‘𝑀)))
27	26	eqeq2d 2746	. . . 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 2735	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
30	29	subm0cl 18738	. . . . 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 18738	. . . . 5 ⊢ (𝐹 ∈ (SubMnd‘𝑀) → (0g‘𝑀) ∈ 𝐹)
34	32, 33	syl 17	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝐹)
35	29	gsum0 18611	. . . . 5 ⊢ (𝑀 Σg ∅) = (0g‘𝑀)
36		gsumwun.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ CMnd)
37	36	cmnmndd 19735	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Mnd)
38		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
39	38, 29	mndidcl 18676	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
40		gsumwun.p	. . . . . . 7 ⊢ + = (+g‘𝑀)
41	38, 40, 29	mndlid 18681	. . . . . 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 2789	. . . 4 ⊢ (𝜑 → (𝑀 Σg ∅) = ((0g‘𝑀) + (0g‘𝑀)))
44	25, 27, 31, 34, 43	2rspcedvdw 3589	. . 3 ⊢ (𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg ∅) = (𝑒 + 𝑓))
45		oveq1 7365	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑒 + 𝑥) → (𝑖 + 𝑗) = ((𝑒 + 𝑥) + 𝑗))
46	45	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑖 = (𝑒 + 𝑥) → ((𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗) ↔ (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = ((𝑒 + 𝑥) + 𝑗)))
47		oveq2 7366	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑓 → ((𝑒 + 𝑥) + 𝑗) = ((𝑒 + 𝑥) + 𝑓))
48	47	eqeq2d 2746	. . . . . . . . . 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 33096	. . . . . . . . . 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 18736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∈ (SubMnd‘𝑀) → 𝐸 ⊆ (Base‘𝑀))
56	28, 55	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ⊆ (Base‘𝑀))
57	38	submss 18736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (SubMnd‘𝑀) → 𝐹 ⊆ (Base‘𝑀))
58	32, 57	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝑀))
59	56, 58	unssd 4143	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ⊆ (Base‘𝑀))
60		sswrd 14447	. . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . 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 18770	. . . . . . . . . . . . 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 7373	. . . . . . . . . . . 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 3932	. . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . 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 19729	. . . . . . . . . . . . . 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 18673	. . . . . . . . . . . 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 2774	. . . . . . . . . . 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 3589	. . . . . . . . 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 7365	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑒 → (𝑖 + 𝑗) = (𝑒 + 𝑗))
85	84	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑖 = 𝑒 → ((𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑖 + 𝑗) ↔ (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = (𝑒 + 𝑗)))
86		oveq2 7366	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑓 + 𝑥) → (𝑒 + 𝑗) = (𝑒 + (𝑓 + 𝑥)))
87	86	eqeq2d 2746	. . . . . . . . . 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 33096	. . . . . . . . . 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 33084	. . . . . . . . . . . 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 2774	. . . . . . . . . . 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 3589	. . . . . . . . 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 4104	. . . . . . . . . . 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 32525	. . . . . . 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 14647	. 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 3059   ∪ cun 3898   ⊆ wss 3900  ∅c0 4284  ‘cfv 6491  (class class class)co 7358  Word cword 14438   ++ cconcat 14495  ⟨“cs1 14521  Basecbs 17138  +_gcplusg 17179  0_gc0g 17361   Σ_g cgsu 17362  Mndcmnd 18661  SubMndcsubmnd 18709  CMndccmn 19711

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-n0 12404  df-xnn0 12477  df-z 12491  df-uz 12754  df-fz 13426  df-fzo 13573  df-seq 13927  df-hash 14256  df-word 14439  df-lsw 14488  df-concat 14496  df-s1 14522  df-substr 14567  df-pfx 14597  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  df-gsum 17364  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-cmn 19713

This theorem is referenced by:  elrgspnsubrunlem2  33309