mrsubccat - Mathbox for Mario Carneiro

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Theorem mrsubccat 35503

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 4346	. . . . . 6 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2		mrsubccat.s	. . . . . . 7 ⊢ 𝑆 = (mRSubst‘𝑇)
3	2	rnfvprc 6901	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
4	1, 3	nsyl2 141	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V)
5		eqid 2735	. . . . . 6 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
6		mrsubccat.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6, 2	mrsubff 35497	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑_pm (mVR‘𝑇))⟶(𝑅 ↑_m 𝑅))
8		ffun 6740	. . . . 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 35498	. . . . . 6 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m (mVR‘𝑇)))
11	10	eleq2i 2831	. . . . 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 6974	. . . 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 771	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑋 ∈ 𝑅)
16		elfvex 6945	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (mREx‘𝑇) → 𝑇 ∈ V)
17	16, 6	eleq2s 2857	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑅 → 𝑇 ∈ V)
18		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
19	18, 5, 6	mrexval 35486	. . . . . . . . . . . . 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 2841	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
22		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑌 ∈ 𝑅)
23	22, 20	eleqtrd 2841	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
24		elmapi 8888	. . . . . . . . . . . . . . . . 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 7104	. . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . . 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 14638	. . . . . . . . . . . . 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 4566	. . . . . . . . . . . 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 7135	. . . . . . . . . . 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 14871	. . . . . . . . . . 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 1370	. . . . . . . . . 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 7447	. . . . . . . . 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 6920	. . . . . . . . . . . 12 ⊢ (mCN‘𝑇) ∈ V
38		fvex 6920	. . . . . . . . . . . 12 ⊢ (mVR‘𝑇) ∈ V
39	37, 38	unex 7763	. . . . . . . . . . 11 ⊢ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V
40		eqid 2735	. . . . . . . . . . . 12 ⊢ (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))
41	40	frmdmnd 18885	. . . . . . . . . . 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 14867	. . . . . . . . . . 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 14867	. . . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
48	40, 47	frmdbas 18878	. . . . . . . . . . . . 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 2744	. . . . . . . . . . 11 ⊢ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
51		eqid 2735	. . . . . . . . . . 11 ⊢ (+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
52	50, 51	gsumccat 18867	. . . . . . . . . 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 1370	. . . . . . . . 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 18865	. . . . . . . . . . 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 18865	. . . . . . . . . . 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 18881	. . . . . . . . . 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 2779	. . . . . . . 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 4019	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (mVR‘𝑇) ⊆ (mVR‘𝑇))
62		ccatcl 14609	. . . . . . . . . . 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 2842	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (𝑋 ++ 𝑌) ∈ 𝑅)
65	18, 5, 6, 2, 40	mrsubval 35494	. . . . . . . . 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 1370	. . . . . . . 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 35494	. . . . . . . . . 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 1370	. . . . . . . . 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 35494	. . . . . . . . . 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 1370	. . . . . . . . 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 7449	. . . . . . . 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 2785	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)))
73		fveq1 6906	. . . . . . . 8 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = (𝐹‘(𝑋 ++ 𝑌)))
74		fveq1 6906	. . . . . . . . 9 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘𝑋) = (𝐹‘𝑋))
75		fveq1 6906	. . . . . . . . 9 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘𝑌) = (𝐹‘𝑌))
76	74, 75	oveq12d 7449	. . . . . . . 8 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌)))
77	73, 76	eqeq12d 2751	. . . . . . 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 3146	. . 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 1115	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 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 ifcif 4531 ↦ cmpt 5231 ran crn 5690 “ cima 5692 ∘ ccom 5693 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 ↑_pm cpm 8866 Word cword 14549 ++ cconcat 14605 ⟨“cs1 14630 Basecbs 17245 +_gcplusg 17298 Σ_g cgsu 17487 Mndcmnd 18760 freeMndcfrmd 18873 mCNcmcn 35445 mVRcmvar 35446 mRExcmrex 35451 mRSubstcmrsub 35455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-frmd 18875 df-mrex 35471 df-mrsub 35475

This theorem is referenced by: elmrsubrn 35505 mrsubco 35506 mrsubvrs 35507

W3C validator