Theorem gsumwun 33005

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 7411	. . . . . 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 3206	. . . 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 7411	. . . . . 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 3206	. . . 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 7410	. . . . . . 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 7411	. . . . . . 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 3225	. . . . 5 ⊢ (∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑒 + 𝑓) ↔ ∃𝑖 ∈ 𝐸 ∃𝑗 ∈ 𝐹 (𝑀 Σg 𝑣) = (𝑖 + 𝑗))
15		oveq2 7411	. . . . . . 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 3206	. . . . 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 7411	. . . . . 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 3206	. . . 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 7410	. . . . 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 7411	. . . . 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 18787	. . . . 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 18787	. . . . 5 ⊢ (𝐹 ∈ (SubMnd‘𝑀) → (0g‘𝑀) ∈ 𝐹)
34	32, 33	syl 17	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝐹)
35	29	gsum0 18660	. . . . 5 ⊢ (𝑀 Σg ∅) = (0g‘𝑀)
36		gsumwun.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ CMnd)
37	36	cmnmndd 19783	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Mnd)
38		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
39	38, 29	mndidcl 18725	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
40		gsumwun.p	. . . . . . 7 ⊢ + = (+g‘𝑀)
41	38, 40, 29	mndlid 18730	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ (0g‘𝑀) ∈ (Base‘𝑀)) → ((0g‘𝑀) + (0g‘𝑀)) = (0g‘𝑀))
42	37, 39, 41	syl2anc2 585	. . . . 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 3615	. . 3 ⊢ (𝜑 → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 (𝑀 Σg ∅) = (𝑒 + 𝑓))
45		oveq1 7410	. . . . . . . . . . 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 7411	. . . . . . . . . . 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 736	. . . . . . . . . . 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 783	. . . . . . . . . . 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 32976	. . . . . . . . . 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 775	. . . . . . . . . 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 734	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → 𝑀 ∈ Mnd)
55	38	submss 18785	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∈ (SubMnd‘𝑀) → 𝐸 ⊆ (Base‘𝑀))
56	28, 55	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ⊆ (Base‘𝑀))
57	38	submss 18785	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (SubMnd‘𝑀) → 𝐹 ⊆ (Base‘𝑀))
58	32, 57	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝑀))
59	56, 58	unssd 4167	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ⊆ (Base‘𝑀))
60		sswrd 14538	. . . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → 𝑤 ∈ Word (Base‘𝑀))
63	62	ad4antr 732	. . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) → 𝑥 ∈ (Base‘𝑀))
66	65	ad3antrrr 730	. . . . . . . . . . . . 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 18819	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ Mnd ∧ 𝑤 ∈ Word (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑀 Σg (𝑤 ++ ⟨“𝑥”⟩)) = ((𝑀 Σg 𝑤) + 𝑥))
68	54, 63, 66, 67	syl3anc 1373	. . . . . . . . . . . 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 7418	. . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) → 𝐸 ⊆ (Base‘𝑀))
72	71	sselda 3958	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ (Base‘𝑀))
73	72	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → 𝑒 ∈ (Base‘𝑀))
74	58	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) → 𝐹 ⊆ (Base‘𝑀))
75	74	sselda 3958	. . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑥 ∈ (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ (𝑀 Σg 𝑤) = (𝑒 + 𝑓)) → 𝑀 ∈ CMnd)
78	38, 40	cmncom 19777	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ CMnd ∧ 𝑓 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑓 + 𝑥) = (𝑥 + 𝑓))
79	77, 76, 66, 78	syl3anc 1373	. . . . . . . . . . . . 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 18722	. . . . . . . . . . . 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 3615	. . . . . . . . 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 7410	. . . . . . . . . . 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 7411	. . . . . . . . . . 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 783	. . . . . . . . . 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 736	. . . . . . . . . . 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 775	. . . . . . . . . . 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 32976	. . . . . . . . . 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 32964	. . . . . . . . . . . 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 3615	. . . . . . . . 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 4128	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐸 ∪ 𝐹) ↔ (𝑥 ∈ 𝐸 ∨ 𝑥 ∈ 𝐹))
98	97	biimpi 216	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐸 ∪ 𝐹) → (𝑥 ∈ 𝐸 ∨ 𝑥 ∈ 𝐹))
99	98	ad4antlr 733	. . . . . . . . 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 960	. . . . . . . 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 32398	. . . . . . 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 14738	. 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 847   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   ∪ cun 3924   ⊆ wss 3926  ∅c0 4308  ‘cfv 6530  (class class class)co 7403  Word cword 14529   ++ cconcat 14586  ⟨“cs1 14611  Basecbs 17226  +_gcplusg 17269  0_gc0g 17451   Σ_g cgsu 17452  Mndcmnd 18710  SubMndcsubmnd 18758  CMndccmn 19759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-nn 12239  df-2 12301  df-n0 12500  df-xnn0 12573  df-z 12587  df-uz 12851  df-fz 13523  df-fzo 13670  df-seq 14018  df-hash 14347  df-word 14530  df-lsw 14579  df-concat 14587  df-s1 14612  df-substr 14657  df-pfx 14687  df-sets 17181  df-slot 17199  df-ndx 17211  df-base 17227  df-ress 17250  df-plusg 17282  df-0g 17453  df-gsum 17454  df-mgm 18616  df-sgrp 18695  df-mnd 18711  df-submnd 18760  df-cmn 19761

This theorem is referenced by:  elrgspnsubrunlem2  33189