Theorem seqof2 13429

Description: Distribute function operation through a sequence. Maps-to notation version of seqof 13428. (Contributed by Mario Carneiro, 7-Jul-2017.)

Hypotheses

Ref	Expression
seqof2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
seqof2.2	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seqof2.3	⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐵)
seqof2.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)) → 𝑋 ∈ 𝑊)

Assertion

Ref	Expression
seqof2	⊢ (𝜑 → (seq𝑀( ∘f + , (𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋)))‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁)))

Distinct variable groups:   𝑥,𝑧,𝐴   𝑥,𝑀,𝑧   𝑥,𝑁,𝑧   𝜑,𝑥,𝑧   𝑧, +   𝑥,𝐵

Allowed substitution hints:   𝐵(𝑧)   + (𝑥)   𝑉(𝑥,𝑧)   𝑊(𝑥,𝑧)   𝑋(𝑥,𝑧)

Proof of Theorem seqof2

Dummy variables 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqof2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		seqof2.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
3		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ (𝑀...𝑁))
4		nffvmpt1 6681	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑛)
5		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
6		nffvmpt1 6681	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛)
7	5, 6	nfmpt 5163	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛))
8	4, 7	nfeq 2991	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑛) = (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛))
9	3, 8	nfim 1897	. . . . 5 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑛) = (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛)))
10		eleq1w 2895	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
11	10	anbi2d 630	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝑛 ∈ (𝑀...𝑁))))
12		fveq2 6670	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑥) = ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑛))
13		fveq2 6670	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛))
14	13	mpteq2dv 5162	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥)) = (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛)))
15	12, 14	eqeq12d 2837	. . . . . 6 ⊢ (𝑥 = 𝑛 → (((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑥) = (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥)) ↔ ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑛) = (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛))))
16	11, 15	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑛 → (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑥) = (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥))) ↔ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑛) = (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛)))))
17		seqof2.3	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐵)
18	17	sselda 3967	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ 𝐵)
19	1	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐴 ∈ 𝑉)
20	19	mptexd 6987	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑧 ∈ 𝐴 ↦ 𝑋) ∈ V)
21		eqid 2821	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋)) = (𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))
22	21	fvmpt2 6779	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ↦ 𝑋) ∈ V) → ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑥) = (𝑧 ∈ 𝐴 ↦ 𝑋))
23	18, 20, 22	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑥) = (𝑧 ∈ 𝐴 ↦ 𝑋))
24	18	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ 𝐵)
25		simpll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ 𝑧 ∈ 𝐴) → 𝜑)
26		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
27		seqof2.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)) → 𝑋 ∈ 𝑊)
28	25, 24, 26, 27	syl12anc 834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ 𝑧 ∈ 𝐴) → 𝑋 ∈ 𝑊)
29		eqid 2821	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ 𝑋) = (𝑥 ∈ 𝐵 ↦ 𝑋)
30	29	fvmpt2 6779	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝑊) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
31	24, 28, 30	syl2anc 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥) = 𝑋)
32	31	mpteq2dva 5161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥)) = (𝑧 ∈ 𝐴 ↦ 𝑋))
33	23, 32	eqtr4d 2859	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑥) = (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑥)))
34	9, 16, 33	chvarfv 2242	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑛) = (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛)))
35		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛)
36		nfcsb1v 3907	. . . . . 6 ⊢ Ⅎ𝑧⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋)
37		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑧𝑛
38	36, 37	nffv 6680	. . . . 5 ⊢ Ⅎ𝑧(⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛)
39		csbeq1a 3897	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝐵 ↦ 𝑋) = ⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋))
40	39	fveq1d 6672	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛) = (⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛))
41	35, 38, 40	cbvmpt 5167	. . . 4 ⊢ (𝑧 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛)) = (𝑦 ∈ 𝐴 ↦ (⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛))
42	34, 41	syl6eq 2872	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → ((𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋))‘𝑛) = (𝑦 ∈ 𝐴 ↦ (⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋)‘𝑛)))
43	1, 2, 42	seqof 13428	. 2 ⊢ (𝜑 → (seq𝑀( ∘f + , (𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋)))‘𝑁) = (𝑦 ∈ 𝐴 ↦ (seq𝑀( + , ⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁)))
44		nfcv 2977	. . 3 ⊢ Ⅎ𝑦(seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁)
45		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑧𝑀
46		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑧 +
47	45, 46, 36	nfseq 13380	. . . 4 ⊢ Ⅎ𝑧seq𝑀( + , ⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋))
48		nfcv 2977	. . . 4 ⊢ Ⅎ𝑧𝑁
49	47, 48	nffv 6680	. . 3 ⊢ Ⅎ𝑧(seq𝑀( + , ⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁)
50	39	seqeq3d 13378	. . . 4 ⊢ (𝑧 = 𝑦 → seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋)) = seq𝑀( + , ⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋)))
51	50	fveq1d 6672	. . 3 ⊢ (𝑧 = 𝑦 → (seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁) = (seq𝑀( + , ⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁))
52	44, 49, 51	cbvmpt 5167	. 2 ⊢ (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁)) = (𝑦 ∈ 𝐴 ↦ (seq𝑀( + , ⦋𝑦 / 𝑧⦌(𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁))
53	43, 52	syl6eqr 2874	1 ⊢ (𝜑 → (seq𝑀( ∘f + , (𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋)))‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ⦋csb 3883   ⊆ wss 3936   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156   ∘_f cof 7407  ℤ_≥cuz 12244  ...cfz 12893  seqcseq 13370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-seq 13371

This theorem is referenced by:  mtestbdd  24993  lgamgulm2  25613  lgamcvglem  25617