mrsubccat - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mrsubccat		Structured version Visualization version GIF version

Theorem mrsubccat 33480

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 4267	. . . . . 6 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2		mrsubccat.s	. . . . . . 7 ⊢ 𝑆 = (mRSubst‘𝑇)
3	2	rnfvprc 6768	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
4	1, 3	nsyl2 141	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V)
5		eqid 2738	. . . . . 6 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
6		mrsubccat.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6, 2	mrsubff 33474	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑_pm (mVR‘𝑇))⟶(𝑅 ↑_m 𝑅))
8		ffun 6603	. . . . 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 33475	. . . . . 6 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑_m (mVR‘𝑇)))
11	10	eleq2i 2830	. . . . 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 6835	. . . 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 768	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → 𝑋 ∈ 𝑅)
16		elfvex 6807	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (mREx‘𝑇) → 𝑇 ∈ V)
17	16, 6	eleq2s 2857	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑅 → 𝑇 ∈ V)
18		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
19	18, 5, 6	mrexval 33463	. . . . . . . . . . . . 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 770	. . . . . . . . . . . 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 8637	. . . . . . . . . . . . . . . . 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	ffvelrnda 6961	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓‘𝑣) ∈ 𝑅)
28	20	ad2antrr 723	. . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . 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 14308	. . . . . . . . . . . . 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 4494	. . . . . . . . . . . 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 6989	. . . . . . . . . . 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 14548	. . . . . . . . . . 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 7291	. . . . . . . . 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 6787	. . . . . . . . . . . 12 ⊢ (mCN‘𝑇) ∈ V
38		fvex 6787	. . . . . . . . . . . 12 ⊢ (mVR‘𝑇) ∈ V
39	37, 38	unex 7596	. . . . . . . . . . 11 ⊢ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V
40		eqid 2738	. . . . . . . . . . . 12 ⊢ (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))
41	40	frmdmnd 18498	. . . . . . . . . . 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 14544	. . . . . . . . . . 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 14544	. . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
48	40, 47	frmdbas 18491	. . . . . . . . . . . . 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 2747	. . . . . . . . . . 11 ⊢ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
51		eqid 2738	. . . . . . . . . . 11 ⊢ (+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (+_g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
52	50, 51	gsumccat 18480	. . . . . . . . . 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 18477	. . . . . . . . . . 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 18477	. . . . . . . . . . 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 18494	. . . . . . . . . 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 2782	. . . . . . . 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 3944	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → (mVR‘𝑇) ⊆ (mVR‘𝑇))
62		ccatcl 14277	. . . . . . . . . . 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 33471	. . . . . . . . 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 33471	. . . . . . . . . 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 33471	. . . . . . . . . 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 7293	. . . . . . . 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 2788	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑅 ↑_m (mVR‘𝑇)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅)) → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)))
73		fveq1 6773	. . . . . . . 8 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘(𝑋 ++ 𝑌)) = (𝐹‘(𝑋 ++ 𝑌)))
74		fveq1 6773	. . . . . . . . 9 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘𝑋) = (𝐹‘𝑋))
75		fveq1 6773	. . . . . . . . 9 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘𝑌) = (𝐹‘𝑌))
76	74, 75	oveq12d 7293	. . . . . . . 8 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘𝑋) ++ ((𝑆‘𝑓)‘𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌)))
77	73, 76	eqeq12d 2754	. . . . . . 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 3209	. . 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 Vcvv 3432 ∪ cun 3885 ⊆ wss 3887 ∅c0 4256 ifcif 4459 ↦ cmpt 5157 ran crn 5590 “ cima 5592 ∘ ccom 5593 Fun wfun 6427 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 ↑_pm cpm 8616 Word cword 14217 ++ cconcat 14273 ⟨“cs1 14300 Basecbs 16912 +_gcplusg 16962 Σ_g cgsu 17151 Mndcmnd 18385 freeMndcfrmd 18486 mCNcmcn 33422 mVRcmvar 33423 mRExcmrex 33428 mRSubstcmrsub 33432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-word 14218 df-concat 14274 df-s1 14301 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-0g 17152 df-gsum 17153 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-frmd 18488 df-mrex 33448 df-mrsub 33452

This theorem is referenced by: elmrsubrn 33482 mrsubco 33483 mrsubvrs 33484

W3C validator