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Theorem fsumcnv 15698

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 3862	. . . 4 ⊢ (𝑥 = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		fvex 6846	. . . . 5 ⊢ (2^nd ‘𝑦) ∈ V
3		fvex 6846	. . . . 5 ⊢ (1^st ‘𝑦) ∈ V
4		opex 5411	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
5		fsumcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
6	4, 5	csbie 3883	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
7		opeq12 4830	. . . . . . 7 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
8	7	csbeq1d 3852	. . . . . 6 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
9	6, 8	eqtr3id 2784	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
10	2, 3, 9	csbie2 3887	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵
11	1, 10	eqtr4di 2788	. . 3 ⊢ (𝑥 = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ → 𝐵 = ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
12		fsumcnv.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		cnvfi 9102	. . . 4 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → ^◡𝐴 ∈ Fin)
15		relcnv 6062	. . . . 5 ⊢ Rel ^◡𝐴
16		cnvf1o 8053	. . . . 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 6146	. . . . . 6 ⊢ (Rel 𝐴 ↔ ^◡^◡𝐴 = 𝐴)
20	18, 19	sylib 218	. . . . 5 ⊢ (𝜑 → ^◡^◡𝐴 = 𝐴)
21	20	f1oeq3d 6770	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}):^◡𝐴–1-1-onto→^◡^◡𝐴 ↔ (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}):^◡𝐴–1-1-onto→𝐴))
22	17, 21	mpbii 233	. . 3 ⊢ (𝜑 → (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}):^◡𝐴–1-1-onto→𝐴)
23		1st2nd 7983	. . . . . . 7 ⊢ ((Rel ^◡𝐴 ∧ 𝑦 ∈ ^◡𝐴) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
24	15, 23	mpan 691	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
25	24	fveq2d 6837	. . . . 5 ⊢ (𝑦 ∈ ^◡𝐴 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘𝑦) = ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩))
26		id 22	. . . . . . 7 ⊢ (𝑦 ∈ ^◡𝐴 → 𝑦 ∈ ^◡𝐴)
27	24, 26	eqeltrrd 2836	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ ^◡𝐴)
28		sneq 4589	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → {𝑧} = {⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
29	28	cnveqd 5823	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ^◡{𝑧} = ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
30	29	unieqd 4875	. . . . . . . 8 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ∪ ^◡{𝑧} = ∪ ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
31		opswap 6186	. . . . . . . 8 ⊢ ∪ ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩
32	30, 31	eqtrdi 2786	. . . . . . 7 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ∪ ^◡{𝑧} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
33		eqid 2735	. . . . . . 7 ⊢ (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}) = (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})
34		opex 5411	. . . . . . 7 ⊢ ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ V
35	32, 33, 34	fvmpt 6940	. . . . . 6 ⊢ (⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ ^◡𝐴 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
36	27, 35	syl 17	. . . . 5 ⊢ (𝑦 ∈ ^◡𝐴 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
37	25, 36	eqtrd 2770	. . . 4 ⊢ (𝑦 ∈ ^◡𝐴 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
38	37	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ^◡𝐴) → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
39		fsumcnv.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
40	11, 14, 22, 38, 39	fsumf1o 15648	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
41		csbeq1a 3862	. . . . 5 ⊢ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → 𝐶 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
42	24, 41	syl 17	. . . 4 ⊢ (𝑦 ∈ ^◡𝐴 → 𝐶 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
43		opex 5411	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
44		fsumcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
45	43, 44	csbie 3883	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
46		opeq12 4830	. . . . . . . 8 ⊢ ((𝑘 = (1^st ‘𝑦) ∧ 𝑗 = (2^nd ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
47	46	ancoms 458	. . . . . . 7 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
48	47	csbeq1d 3852	. . . . . 6 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
49	45, 48	eqtr3id 2784	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
50	2, 3, 49	csbie2 3887	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶
51	42, 50	eqtr4di 2788	. . 3 ⊢ (𝑦 ∈ ^◡𝐴 → 𝐶 = ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
52	51	sumeq2i 15623	. 2 ⊢ Σ𝑦 ∈ ^◡ 𝐴𝐶 = Σ𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷
53	40, 52	eqtr4di 2788	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ^◡ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⦋csb 3848 {csn 4579 ⟨cop 4585 ∪ cuni 4862 ↦ cmpt 5178 ^◡ccnv 5622 Rel wrel 5628 –1-1-onto→wf1o 6490 ‘cfv 6491 1^st c1st 7931 2^nd c2nd 7932 Fincfn 8885 ℂcc 11026 Σcsu 15611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-sup 9347 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12754 df-rp 12908 df-fz 13426 df-fzo 13573 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612

This theorem is referenced by: fsumcom2 15699

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