Theorem fsumcnv 11948

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 3133	. . . 4 ⊢ (𝑥 = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		2ndexg 6314	. . . . . 6 ⊢ (𝑦 ∈ V → (2nd ‘𝑦) ∈ V)
3	2	elv 2803	. . . . 5 ⊢ (2nd ‘𝑦) ∈ V
4		1stexg 6313	. . . . . 6 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
5	4	elv 2803	. . . . 5 ⊢ (1st ‘𝑦) ∈ V
6		vex 2802	. . . . . . . 8 ⊢ 𝑗 ∈ V
7		vex 2802	. . . . . . . 8 ⊢ 𝑘 ∈ V
8	6, 7	opex 4315	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
9		fsumcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
10	8, 9	csbie 3170	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
11		opeq12 3859	. . . . . . 7 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
12	11	csbeq1d 3131	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
13	10, 12	eqtr3id 2276	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
14	3, 5, 13	csbie2 3174	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵
15	1, 14	eqtr4di 2280	. . 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 7108	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin)
19	16, 17, 18	syl2anc 411	. . 3 ⊢ (𝜑 → ◡𝐴 ∈ Fin)
20		relcnv 5106	. . . . 5 ⊢ Rel ◡𝐴
21		cnvf1o 6371	. . . . 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 5179	. . . . . 6 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
24	16, 23	sylib 122	. . . . 5 ⊢ (𝜑 → ◡◡𝐴 = 𝐴)
25		f1oeq3 5562	. . . . 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 148	. . 3 ⊢ (𝜑 → (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→𝐴)
28		1st2nd 6327	. . . . . . 7 ⊢ ((Rel ◡𝐴 ∧ 𝑦 ∈ ◡𝐴) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
29	20, 28	mpan 424	. . . . . 6 ⊢ (𝑦 ∈ ◡𝐴 → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
30	29	fveq2d 5631	. . . . 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 2307	. . . . . 6 ⊢ (𝑦 ∈ ◡𝐴 → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ◡𝐴)
33		sneq 3677	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → {𝑧} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
34	33	cnveqd 4898	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ◡{𝑧} = ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
35	34	unieqd 3899	. . . . . . . 8 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
36		opswapg 5215	. . . . . . . . 9 ⊢ (((1st ‘𝑦) ∈ V ∧ (2nd ‘𝑦) ∈ V) → ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
37	5, 3, 36	mp2an 426	. . . . . . . 8 ⊢ ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩
38	35, 37	eqtrdi 2278	. . . . . . 7 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
39		eqid 2229	. . . . . . 7 ⊢ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}) = (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})
40	3, 5	opex 4315	. . . . . . 7 ⊢ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ ∈ V
41	38, 39, 40	fvmpt 5711	. . . . . 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 2262	. . . 4 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
44	43	adantl 277	. . 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 11901	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
47		csbeq1a 3133	. . . . 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 4315	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
50		fsumcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
51	49, 50	csbie 3170	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
52		opeq12 3859	. . . . . . . 8 ⊢ ((𝑘 = (1st ‘𝑦) ∧ 𝑗 = (2nd ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
53	52	ancoms 268	. . . . . . 7 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
54	53	csbeq1d 3131	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
55	51, 54	eqtr3id 2276	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
56	3, 5, 55	csbie2 3174	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶
57	48, 56	eqtr4di 2280	. . 3 ⊢ (𝑦 ∈ ◡𝐴 → 𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
58	57	sumeq2i 11875	. 2 ⊢ Σ𝑦 ∈ ◡ 𝐴𝐶 = Σ𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷
59	46, 58	eqtr4di 2280	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ⦋csb 3124  {csn 3666  ⟨cop 3669  ∪ cuni 3888   ↦ cmpt 4145  ^◡ccnv 4718  Rel wrel 4724  –1-1-onto→wf1o 5317  ‘cfv 5318  1^st c1st 6284  2^nd c2nd 6285  Fincfn 6887  ℂcc 7997  Σcsu 11864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865

This theorem is referenced by:  fisumcom2  11949