Theorem fsumcnv 11400

Description: Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
fsumcnv.1	⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
fsumcnv.2	⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
fsumcnv.3	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumcnv.4	⊢ (𝜑 → Rel 𝐴)
fsumcnv.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
fsumcnv	⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑗,𝑘,𝑦,𝐵   𝑥,𝑗,𝐶,𝑘   𝜑,𝑥,𝑦   𝑥,𝐷,𝑦

Allowed substitution hints:   𝜑(𝑗,𝑘)   𝐴(𝑗,𝑘)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑗,𝑘)

Proof of Theorem fsumcnv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1a 3058	. . . 4 ⊢ (𝑥 = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		2ndexg 6147	. . . . . 6 ⊢ (𝑦 ∈ V → (2nd ‘𝑦) ∈ V)
3	2	elv 2734	. . . . 5 ⊢ (2nd ‘𝑦) ∈ V
4		1stexg 6146	. . . . . 6 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
5	4	elv 2734	. . . . 5 ⊢ (1st ‘𝑦) ∈ V
6		vex 2733	. . . . . . . 8 ⊢ 𝑗 ∈ V
7		vex 2733	. . . . . . . 8 ⊢ 𝑘 ∈ V
8	6, 7	opex 4214	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
9		fsumcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
10	8, 9	csbie 3094	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
11		opeq12 3767	. . . . . . 7 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
12	11	csbeq1d 3056	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
13	10, 12	eqtr3id 2217	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
14	3, 5, 13	csbie2 3098	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵
15	1, 14	eqtr4di 2221	. . 3 ⊢ (𝑥 = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ → 𝐵 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
16		fsumcnv.4	. . . 4 ⊢ (𝜑 → Rel 𝐴)
17		fsumcnv.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
18		relcnvfi 6918	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin)
19	16, 17, 18	syl2anc 409	. . 3 ⊢ (𝜑 → ◡𝐴 ∈ Fin)
20		relcnv 4989	. . . . 5 ⊢ Rel ◡𝐴
21		cnvf1o 6204	. . . . 5 ⊢ (Rel ◡𝐴 → (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴)
22	20, 21	ax-mp 5	. . . 4 ⊢ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴
23		dfrel2 5061	. . . . . 6 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
24	16, 23	sylib 121	. . . . 5 ⊢ (𝜑 → ◡◡𝐴 = 𝐴)
25		f1oeq3 5433	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴 ↔ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→𝐴))
26	24, 25	syl 14	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴 ↔ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→𝐴))
27	22, 26	mpbii 147	. . 3 ⊢ (𝜑 → (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→𝐴)
28		1st2nd 6160	. . . . . . 7 ⊢ ((Rel ◡𝐴 ∧ 𝑦 ∈ ◡𝐴) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
29	20, 28	mpan 422	. . . . . 6 ⊢ (𝑦 ∈ ◡𝐴 → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
30	29	fveq2d 5500	. . . . 5 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩))
31		id 19	. . . . . . 7 ⊢ (𝑦 ∈ ◡𝐴 → 𝑦 ∈ ◡𝐴)
32	29, 31	eqeltrrd 2248	. . . . . 6 ⊢ (𝑦 ∈ ◡𝐴 → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ◡𝐴)
33		sneq 3594	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → {𝑧} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
34	33	cnveqd 4787	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ◡{𝑧} = ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
35	34	unieqd 3807	. . . . . . . 8 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
36		opswapg 5097	. . . . . . . . 9 ⊢ (((1st ‘𝑦) ∈ V ∧ (2nd ‘𝑦) ∈ V) → ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
37	5, 3, 36	mp2an 424	. . . . . . . 8 ⊢ ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩
38	35, 37	eqtrdi 2219	. . . . . . 7 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
39		eqid 2170	. . . . . . 7 ⊢ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}) = (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})
40	3, 5	opex 4214	. . . . . . 7 ⊢ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ ∈ V
41	38, 39, 40	fvmpt 5573	. . . . . 6 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
42	32, 41	syl 14	. . . . 5 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
43	30, 42	eqtrd 2203	. . . 4 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
44	43	adantl 275	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ◡𝐴) → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
45		fsumcnv.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
46	15, 19, 27, 44, 45	fsumf1o 11353	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
47		csbeq1a 3058	. . . . 5 ⊢ (𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → 𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
48	29, 47	syl 14	. . . 4 ⊢ (𝑦 ∈ ◡𝐴 → 𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
49	7, 6	opex 4214	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
50		fsumcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
51	49, 50	csbie 3094	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
52		opeq12 3767	. . . . . . . 8 ⊢ ((𝑘 = (1st ‘𝑦) ∧ 𝑗 = (2nd ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
53	52	ancoms 266	. . . . . . 7 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
54	53	csbeq1d 3056	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
55	51, 54	eqtr3id 2217	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
56	3, 5, 55	csbie2 3098	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶
57	48, 56	eqtr4di 2221	. . 3 ⊢ (𝑦 ∈ ◡𝐴 → 𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
58	57	sumeq2i 11327	. 2 ⊢ Σ𝑦 ∈ ◡ 𝐴𝐶 = Σ𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷
59	46, 58	eqtr4di 2221	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  Vcvv 2730  ⦋csb 3049  {csn 3583  ⟨cop 3586  ∪ cuni 3796   ↦ cmpt 4050  ^◡ccnv 4610  Rel wrel 4616  –1-1-onto→wf1o 5197  ‘cfv 5198  1^st c1st 6117  2^nd c2nd 6118  Fincfn 6718  ℂcc 7772  Σcsu 11316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by:  fisumcom2  11401