Theorem fsumcnv 11444

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 3066	. . . 4 ⊢ (𝑥 = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		2ndexg 6168	. . . . . 6 ⊢ (𝑦 ∈ V → (2nd ‘𝑦) ∈ V)
3	2	elv 2741	. . . . 5 ⊢ (2nd ‘𝑦) ∈ V
4		1stexg 6167	. . . . . 6 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
5	4	elv 2741	. . . . 5 ⊢ (1st ‘𝑦) ∈ V
6		vex 2740	. . . . . . . 8 ⊢ 𝑗 ∈ V
7		vex 2740	. . . . . . . 8 ⊢ 𝑘 ∈ V
8	6, 7	opex 4229	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
9		fsumcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
10	8, 9	csbie 3102	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
11		opeq12 3780	. . . . . . 7 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
12	11	csbeq1d 3064	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
13	10, 12	eqtr3id 2224	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
14	3, 5, 13	csbie2 3106	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵
15	1, 14	eqtr4di 2228	. . 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 6939	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin)
19	16, 17, 18	syl2anc 411	. . 3 ⊢ (𝜑 → ◡𝐴 ∈ Fin)
20		relcnv 5006	. . . . 5 ⊢ Rel ◡𝐴
21		cnvf1o 6225	. . . . 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 5079	. . . . . 6 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
24	16, 23	sylib 122	. . . . 5 ⊢ (𝜑 → ◡◡𝐴 = 𝐴)
25		f1oeq3 5451	. . . . 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 6181	. . . . . . 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 5519	. . . . 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 2255	. . . . . 6 ⊢ (𝑦 ∈ ◡𝐴 → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ◡𝐴)
33		sneq 3603	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → {𝑧} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
34	33	cnveqd 4803	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ◡{𝑧} = ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
35	34	unieqd 3820	. . . . . . . 8 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
36		opswapg 5115	. . . . . . . . 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 2226	. . . . . . 7 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
39		eqid 2177	. . . . . . 7 ⊢ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}) = (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})
40	3, 5	opex 4229	. . . . . . 7 ⊢ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ ∈ V
41	38, 39, 40	fvmpt 5593	. . . . . 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 2210	. . . 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 11397	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
47		csbeq1a 3066	. . . . 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 4229	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
50		fsumcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
51	49, 50	csbie 3102	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
52		opeq12 3780	. . . . . . . 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 3064	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
55	51, 54	eqtr3id 2224	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
56	3, 5, 55	csbie2 3106	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶
57	48, 56	eqtr4di 2228	. . 3 ⊢ (𝑦 ∈ ◡𝐴 → 𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
58	57	sumeq2i 11371	. 2 ⊢ Σ𝑦 ∈ ◡ 𝐴𝐶 = Σ𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷
59	46, 58	eqtr4di 2228	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  Vcvv 2737  ⦋csb 3057  {csn 3592  ⟨cop 3595  ∪ cuni 3809   ↦ cmpt 4064  ^◡ccnv 4625  Rel wrel 4631  –1-1-onto→wf1o 5215  ‘cfv 5216  1^st c1st 6138  2^nd c2nd 6139  Fincfn 6739  ℂcc 7808  Σcsu 11360

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-fz 10008  df-fzo 10142  df-seqfrec 10445  df-exp 10519  df-ihash 10755  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-clim 11286  df-sumdc 11361

This theorem is referenced by:  fisumcom2  11445