mrsubccat - Mathbox for Mario Carneiro

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Theorem mrsubccat 35510

Description: Substitution distributes over concatenation. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
mrsubccat.s	⊢ 𝑆 = (mRSubst‘𝑇)
mrsubccat.r	⊢ 𝑅 = (mREx‘𝑇)

**Assertion**
Ref	Expression
mrsubccat	⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌)))

Proof of Theorem mrsubccat Dummy variables 𝑓 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 4293	. . . . . 6 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2		mrsubccat.s	. . . . . . 7 ⊢ 𝑆 = (mRSubst‘𝑇)
3	2	rnfvprc 6820	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
4	1, 3	nsyl2 141	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V)
5		eqid 2729	. . . . . 6 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
6		mrsubccat.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6, 2	mrsubff 35504	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑_pm (mVR‘𝑇))⟶(𝑅 ↑_m 𝑅))
8		ffun 6659	. . . . 5 ⊢ (𝑆:(𝑅 ↑_pm (mVR‘𝑇))⟶(𝑅 ↑_m 𝑅) → Fun 𝑆)
9	4, 7, 8	3syl 18	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 → Fun 𝑆)
10	5, 6, 2	mrsubrn 35505	. . . . . 6 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m (mVR‘𝑇)))
11	10	eleq2i 2820	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ (𝑅 ↑_m (mVR‘𝑇))))
12	11	biimpi 216	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ (𝑅 ↑_m (mVR‘𝑇))))
13		fvelima 6892	. . . 4 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ (𝑅 ↑_m (mVR‘𝑇)))) → ∃𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇))(𝑆‘𝑓) = 𝐹)
14	9, 12, 13	syl2anc 584	. . 3 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇))(𝑆‘𝑓) = 𝐹)
15		simprl 770	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑋 ∈ 𝑅)
16		elfvex 6862	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (mREx‘𝑇) → 𝑇 ∈ V)
17	16, 6	eleq2s 2846	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑅 → 𝑇 ∈ V)
18		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
19	18, 5, 6	mrexval 35493	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
20	15, 17, 19	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑅 = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
21	15, 20	eleqtrd 2830	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
22		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑌 ∈ 𝑅)
23	22, 20	eleqtrd 2830	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
24		elmapi 8783	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) → 𝑓:(mVR‘𝑇)⟶𝑅)
25	24	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑓:(mVR‘𝑇)⟶𝑅)
26	25	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → 𝑓:(mVR‘𝑇)⟶𝑅)
27	26	ffvelcdmda 7022	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓‘𝑣) ∈ 𝑅)
28	20	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → 𝑅 = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
29	27, 28	eleqtrd 2830	. . . . . . . . . . . . 13 ⊢ ((((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓‘𝑣) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
30		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
31	30	s1cld 14529	. . . . . . . . . . . . 13 ⊢ ((((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
32	29, 31	ifclda 4514	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
33	32	fmpttd 7053	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ (mVR‘𝑇))⟶Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
34		ccatco 14761	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ (𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ (mVR‘𝑇))⟶Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌)) = (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)))
35	21, 23, 33, 34	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌)) = (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)))
36	35	oveq2d 7369	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌))) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
37		fvex 6839	. . . . . . . . . . . 12 ⊢ (mCN‘𝑇) ∈ V
38		fvex 6839	. . . . . . . . . . . 12 ⊢ (mVR‘𝑇) ∈ V
39	37, 38	unex 7684	. . . . . . . . . . 11 ⊢ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V
40		eqid 2729	. . . . . . . . . . . 12 ⊢ (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))
41	40	frmdmnd 18752	. . . . . . . . . . 11 ⊢ (((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V → (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd)
42	39, 41	mp1i 13	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd)
43		wrdco 14757	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ (𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ (mVR‘𝑇))⟶Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
44	21, 33, 43	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
45		wrdco 14757	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ (𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ (mVR‘𝑇))⟶Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
46	23, 33, 45	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
47		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
48	40, 47	frmdbas 18745	. . . . . . . . . . . . 13 ⊢ (((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V → (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
49	39, 48	ax-mp 5	. . . . . . . . . . . 12 ⊢ (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇))
50	49	eqcomi 2738	. . . . . . . . . . 11 ⊢ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
51		eqid 2729	. . . . . . . . . . 11 ⊢ (+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
52	50, 51	gsumccat 18734	. . . . . . . . . 10 ⊢ (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))(+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
53	42, 44, 46, 52	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))(+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
54	50	gsumwcl 18732	. . . . . . . . . . 11 ⊢ (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
55	42, 44, 54	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
56	50	gsumwcl 18732	. . . . . . . . . . 11 ⊢ (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
57	42, 46, 56	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
58	40, 50, 51	frmdadd 18748	. . . . . . . . . 10 ⊢ ((((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))(+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ++ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
59	55, 57, 58	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))(+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ++ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
60	36, 53, 59	3eqtrd 2768	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ++ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
61		ssidd 3961	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (mVR‘𝑇) ⊆ (mVR‘𝑇))
62		ccatcl 14500	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
63	21, 23, 62	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
64	63, 20	eleqtrrd 2831	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (𝑋 ++ 𝑌) ∈ 𝑅)
65	18, 5, 6, 2, 40	mrsubval 35501	. . . . . . . . 9 ⊢ ((𝑓:(mVR‘𝑇)⟶𝑅 ∧ (mVR‘𝑇) ⊆ (mVR‘𝑇) ∧ (𝑋 ++ 𝑌) ∈ 𝑅) → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌))))
66	25, 61, 64, 65	syl3anc 1373	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌))))
67	18, 5, 6, 2, 40	mrsubval 35501	. . . . . . . . . 10 ⊢ ((𝑓:(mVR‘𝑇)⟶𝑅 ∧ (mVR‘𝑇) ⊆ (mVR‘𝑇) ∧ 𝑋 ∈ 𝑅) → ((𝑆‘𝑓)‘𝑋) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
68	25, 61, 15, 67	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑆‘𝑓)‘𝑋) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
69	18, 5, 6, 2, 40	mrsubval 35501	. . . . . . . . . 10 ⊢ ((𝑓:(mVR‘𝑇)⟶𝑅 ∧ (mVR‘𝑇) ⊆ (mVR‘𝑇) ∧ 𝑌 ∈ 𝑅) → ((𝑆‘𝑓)‘𝑌) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)))
70	25, 61, 22, 69	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑆‘𝑓)‘𝑌) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)))
71	68, 70	oveq12d 7371	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ++ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σ_g ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
72	60, 66, 71	3eqtr4d 2774	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)))
73		fveq1 6825	. . . . . . . 8 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = (𝐹‘(𝑋 ++ 𝑌)))
74		fveq1 6825	. . . . . . . . 9 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘𝑋) = (𝐹‘𝑋))
75		fveq1 6825	. . . . . . . . 9 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘𝑌) = (𝐹‘𝑌))
76	74, 75	oveq12d 7371	. . . . . . . 8 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌)))
77	73, 76	eqeq12d 2745	. . . . . . 7 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)) ↔ (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌))))
78	72, 77	syl5ibcom 245	. . . . . 6 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌))))
79	78	ex 412	. . . . 5 ⊢ (𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) → ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌)))))
80	79	com23 86	. . . 4 ⊢ (𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) → ((𝑆‘𝑓) = 𝐹 → ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌)))))
81	80	rexlimiv 3123	. . 3 ⊢ (∃𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇))(𝑆‘𝑓) = 𝐹 → ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌))))
82	14, 81	syl 17	. 2 ⊢ (𝐹 ∈ ran 𝑆 → ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌))))
83	82	3impib 1116	1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 Vcvv 3438 ∪ cun 3903 ⊆ wss 3905 ∅c0 4286 ifcif 4478 ↦ cmpt 5176 ran crn 5624 “ cima 5626 ∘ ccom 5627 Fun wfun 6480 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ↑_m cmap 8760 ↑_pm cpm 8761 Word cword 14439 ++ cconcat 14496 ⟨“cs1 14521 Basecbs 17139 +_gcplusg 17180 Σ_g cgsu 17363 Mndcmnd 18627 freeMndcfrmd 18740 mCNcmcn 35452 mVRcmvar 35453 mRExcmrex 35458 mRSubstcmrsub 35462

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 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 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 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 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-pm 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-card 9854 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-z 12491 df-uz 12755 df-fz 13430 df-fzo 13577 df-seq 13928 df-hash 14257 df-word 14440 df-concat 14497 df-s1 14522 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-0g 17364 df-gsum 17365 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-submnd 18677 df-frmd 18742 df-mrex 35478 df-mrsub 35482

This theorem is referenced by: elmrsubrn 35512 mrsubco 35513 mrsubvrs 35514

W3C validator