mrsubccat - Mathbox for Mario Carneiro

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Theorem mrsubccat 34504

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 4333	. . . . . 6 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2		mrsubccat.s	. . . . . . 7 ⊢ 𝑆 = (mRSubst‘𝑇)
3	2	rnfvprc 6885	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
4	1, 3	nsyl2 141	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V)
5		eqid 2732	. . . . . 6 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
6		mrsubccat.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6, 2	mrsubff 34498	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑_pm (mVR‘𝑇))⟶(𝑅 ↑_m 𝑅))
8		ffun 6720	. . . . 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 34499	. . . . . 6 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m (mVR‘𝑇)))
11	10	eleq2i 2825	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ (𝑅 ↑_m (mVR‘𝑇))))
12	11	biimpi 215	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ (𝑅 ↑_m (mVR‘𝑇))))
13		fvelima 6957	. . . 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 769	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑋 ∈ 𝑅)
16		elfvex 6929	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (mREx‘𝑇) → 𝑇 ∈ V)
17	16, 6	eleq2s 2851	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑅 → 𝑇 ∈ V)
18		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
19	18, 5, 6	mrexval 34487	. . . . . . . . . . . . 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 2835	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
22		simprr 771	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑌 ∈ 𝑅)
23	22, 20	eleqtrd 2835	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
24		elmapi 8842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) → 𝑓:(mVR‘𝑇)⟶𝑅)
25	24	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑓:(mVR‘𝑇)⟶𝑅)
26	25	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → 𝑓:(mVR‘𝑇)⟶𝑅)
27	26	ffvelcdmda 7086	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓‘𝑣) ∈ 𝑅)
28	20	ad2antrr 724	. . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . 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 767	. . . . . . . . . . . . . 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 14552	. . . . . . . . . . . . 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 4563	. . . . . . . . . . . 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 7114	. . . . . . . . . . 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 14785	. . . . . . . . . . 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 1371	. . . . . . . . . 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 7424	. . . . . . . . 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 6904	. . . . . . . . . . . 12 ⊢ (mCN‘𝑇) ∈ V
38		fvex 6904	. . . . . . . . . . . 12 ⊢ (mVR‘𝑇) ∈ V
39	37, 38	unex 7732	. . . . . . . . . . 11 ⊢ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V
40		eqid 2732	. . . . . . . . . . . 12 ⊢ (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))
41	40	frmdmnd 18739	. . . . . . . . . . 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 14781	. . . . . . . . . . 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 14781	. . . . . . . . . . 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 2732	. . . . . . . . . . . . . 14 ⊢ (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
48	40, 47	frmdbas 18732	. . . . . . . . . . . . 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 2741	. . . . . . . . . . 11 ⊢ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
51		eqid 2732	. . . . . . . . . . 11 ⊢ (+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
52	50, 51	gsumccat 18721	. . . . . . . . . 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 1371	. . . . . . . . 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 18719	. . . . . . . . . . 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 18719	. . . . . . . . . . 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 18735	. . . . . . . . . 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 2776	. . . . . . . 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 4005	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (mVR‘𝑇) ⊆ (mVR‘𝑇))
62		ccatcl 14523	. . . . . . . . . . 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 2836	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (𝑋 ++ 𝑌) ∈ 𝑅)
65	18, 5, 6, 2, 40	mrsubval 34495	. . . . . . . . 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 1371	. . . . . . . 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 34495	. . . . . . . . . 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 1371	. . . . . . . . 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 34495	. . . . . . . . . 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 1371	. . . . . . . . 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 7426	. . . . . . . 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 2782	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)))
73		fveq1 6890	. . . . . . . 8 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = (𝐹‘(𝑋 ++ 𝑌)))
74		fveq1 6890	. . . . . . . . 9 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘𝑋) = (𝐹‘𝑋))
75		fveq1 6890	. . . . . . . . 9 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘𝑌) = (𝐹‘𝑌))
76	74, 75	oveq12d 7426	. . . . . . . 8 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌)))
77	73, 76	eqeq12d 2748	. . . . . . 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 244	. . . . . 6 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌))))
79	78	ex 413	. . . . 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 3148	. . 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 Vcvv 3474 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 ifcif 4528 ↦ cmpt 5231 ran crn 5677 “ cima 5679 ∘ ccom 5680 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ↑_m cmap 8819 ↑_pm cpm 8820 Word cword 14463 ++ cconcat 14519 ⟨“cs1 14544 Basecbs 17143 +_gcplusg 17196 Σ_g cgsu 17385 Mndcmnd 18624 freeMndcfrmd 18727 mCNcmcn 34446 mVRcmvar 34447 mRExcmrex 34452 mRSubstcmrsub 34456

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-word 14464 df-concat 14520 df-s1 14545 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-0g 17386 df-gsum 17387 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-frmd 18729 df-mrex 34472 df-mrsub 34476

This theorem is referenced by: elmrsubrn 34506 mrsubco 34507 mrsubvrs 34508

W3C validator