Theorem fsumcnv 14790

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 3699	. . . 4 ⊢ (𝑥 = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		fvex 6387	. . . . 5 ⊢ (2nd ‘𝑦) ∈ V
3		fvex 6387	. . . . 5 ⊢ (1st ‘𝑦) ∈ V
4		opex 5087	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
5		fsumcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
6	4, 5	csbie 3716	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
7		opeq12 4560	. . . . . . 7 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
8	7	csbeq1d 3697	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
9	6, 8	syl5eqr 2812	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
10	2, 3, 9	csbie2 3720	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵
11	1, 10	syl6eqr 2816	. . 3 ⊢ (𝑥 = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ → 𝐵 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
12		fsumcnv.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		cnvfi 8454	. . . 4 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → ◡𝐴 ∈ Fin)
15		relcnv 5684	. . . . 5 ⊢ Rel ◡𝐴
16		cnvf1o 7477	. . . . 5 ⊢ (Rel ◡𝐴 → (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴)
17	15, 16	ax-mp 5	. . . 4 ⊢ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴
18		fsumcnv.4	. . . . . 6 ⊢ (𝜑 → Rel 𝐴)
19		dfrel2 5765	. . . . . 6 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
20	18, 19	sylib 209	. . . . 5 ⊢ (𝜑 → ◡◡𝐴 = 𝐴)
21		f1oeq3 6311	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴 ↔ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→𝐴))
22	20, 21	syl 17	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴 ↔ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→𝐴))
23	17, 22	mpbii 224	. . 3 ⊢ (𝜑 → (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→𝐴)
24		1st2nd 7413	. . . . . . 7 ⊢ ((Rel ◡𝐴 ∧ 𝑦 ∈ ◡𝐴) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
25	15, 24	mpan 681	. . . . . 6 ⊢ (𝑦 ∈ ◡𝐴 → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
26	25	fveq2d 6378	. . . . 5 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩))
27		id 22	. . . . . . 7 ⊢ (𝑦 ∈ ◡𝐴 → 𝑦 ∈ ◡𝐴)
28	25, 27	eqeltrrd 2844	. . . . . 6 ⊢ (𝑦 ∈ ◡𝐴 → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ◡𝐴)
29		sneq 4343	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → {𝑧} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
30	29	cnveqd 5465	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ◡{𝑧} = ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
31	30	unieqd 4603	. . . . . . . 8 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
32		opswap 5807	. . . . . . . 8 ⊢ ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩
33	31, 32	syl6eq 2814	. . . . . . 7 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
34		eqid 2764	. . . . . . 7 ⊢ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}) = (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})
35		opex 5087	. . . . . . 7 ⊢ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ ∈ V
36	33, 34, 35	fvmpt 6470	. . . . . 6 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
37	28, 36	syl 17	. . . . 5 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
38	26, 37	eqtrd 2798	. . . 4 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
39	38	adantl 473	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ◡𝐴) → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
40		fsumcnv.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
41	11, 14, 23, 39, 40	fsumf1o 14740	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
42		csbeq1a 3699	. . . . 5 ⊢ (𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → 𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
43	25, 42	syl 17	. . . 4 ⊢ (𝑦 ∈ ◡𝐴 → 𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
44		opex 5087	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
45		fsumcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
46	44, 45	csbie 3716	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
47		opeq12 4560	. . . . . . . 8 ⊢ ((𝑘 = (1st ‘𝑦) ∧ 𝑗 = (2nd ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
48	47	ancoms 450	. . . . . . 7 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
49	48	csbeq1d 3697	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
50	46, 49	syl5eqr 2812	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
51	2, 3, 50	csbie2 3720	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶
52	43, 51	syl6eqr 2816	. . 3 ⊢ (𝑦 ∈ ◡𝐴 → 𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
53	52	sumeq2i 14715	. 2 ⊢ Σ𝑦 ∈ ◡ 𝐴𝐶 = Σ𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷
54	41, 53	syl6eqr 2816	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ◡ 𝐴𝐶)

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1652   ∈ wcel 2155  ⦋csb 3690  {csn 4333  ⟨cop 4339  ∪ cuni 4593   ↦ cmpt 4887  ^◡ccnv 5275  Rel wrel 5281  –1-1-onto→wf1o 6066  ‘cfv 6067  1^st c1st 7363  2^nd c2nd 7364  Fincfn 8159  ℂcc 10186  Σcsu 14702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265  ax-pre-sup 10266

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-se 5236  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-isom 6076  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-sup 8554  df-oi 8621  df-card 9015  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-div 10938  df-nn 11274  df-2 11334  df-3 11335  df-n0 11538  df-z 11624  df-uz 11886  df-rp 12028  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14125  df-re 14126  df-im 14127  df-sqrt 14261  df-abs 14262  df-clim 14505  df-sum 14703

This theorem is referenced by:  fsumcom2  14791