fsumcnv - Metamath Proof Explorer

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Theorem fsumcnv 15722

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 3902	. . . 4 ⊢ (𝑥 = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		fvex 6897	. . . . 5 ⊢ (2^nd ‘𝑦) ∈ V
3		fvex 6897	. . . . 5 ⊢ (1^st ‘𝑦) ∈ V
4		opex 5457	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
5		fsumcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
6	4, 5	csbie 3924	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
7		opeq12 4870	. . . . . . 7 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
8	7	csbeq1d 3892	. . . . . 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 2780	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
10	2, 3, 9	csbie2 3930	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵
11	1, 10	eqtr4di 2784	. . 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 9179	. . . 4 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → ^◡𝐴 ∈ Fin)
15		relcnv 6096	. . . . 5 ⊢ Rel ^◡𝐴
16		cnvf1o 8094	. . . . 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 6181	. . . . . 6 ⊢ (Rel 𝐴 ↔ ^◡^◡𝐴 = 𝐴)
20	18, 19	sylib 217	. . . . 5 ⊢ (𝜑 → ^◡^◡𝐴 = 𝐴)
21	20	f1oeq3d 6823	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}):^◡𝐴–1-1-onto→^◡^◡𝐴 ↔ (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}):^◡𝐴–1-1-onto→𝐴))
22	17, 21	mpbii 232	. . 3 ⊢ (𝜑 → (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}):^◡𝐴–1-1-onto→𝐴)
23		1st2nd 8021	. . . . . . 7 ⊢ ((Rel ^◡𝐴 ∧ 𝑦 ∈ ^◡𝐴) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
24	15, 23	mpan 687	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
25	24	fveq2d 6888	. . . . 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 2828	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ ^◡𝐴)
28		sneq 4633	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → {𝑧} = {⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
29	28	cnveqd 5868	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ^◡{𝑧} = ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
30	29	unieqd 4915	. . . . . . . 8 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ∪ ^◡{𝑧} = ∪ ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
31		opswap 6221	. . . . . . . 8 ⊢ ∪ ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩
32	30, 31	eqtrdi 2782	. . . . . . 7 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ∪ ^◡{𝑧} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
33		eqid 2726	. . . . . . 7 ⊢ (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}) = (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})
34		opex 5457	. . . . . . 7 ⊢ ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ V
35	32, 33, 34	fvmpt 6991	. . . . . 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 2766	. . . 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 15672	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
41		csbeq1a 3902	. . . . 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 5457	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
44		fsumcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
45	43, 44	csbie 3924	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
46		opeq12 4870	. . . . . . . 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 3892	. . . . . 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 2780	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
50	2, 3, 49	csbie2 3930	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶
51	42, 50	eqtr4di 2784	. . 3 ⊢ (𝑦 ∈ ^◡𝐴 → 𝐶 = ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
52	51	sumeq2i 15648	. 2 ⊢ Σ𝑦 ∈ ^◡ 𝐴𝐶 = Σ𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷
53	40, 52	eqtr4di 2784	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ^◡ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⦋csb 3888 {csn 4623 ⟨cop 4629 ∪ cuni 4902 ↦ cmpt 5224 ^◡ccnv 5668 Rel wrel 5674 –1-1-onto→wf1o 6535 ‘cfv 6536 1^st c1st 7969 2^nd c2nd 7970 Fincfn 8938 ℂcc 11107 Σcsu 15635

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-sum 15636

This theorem is referenced by: fsumcom2 15723

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