fsumcnv - Metamath Proof Explorer

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Theorem fsumcnv 15809

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 3913	. . . 4 ⊢ (𝑥 = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		fvex 6919	. . . . 5 ⊢ (2^nd ‘𝑦) ∈ V
3		fvex 6919	. . . . 5 ⊢ (1^st ‘𝑦) ∈ V
4		opex 5469	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
5		fsumcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
6	4, 5	csbie 3934	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
7		opeq12 4875	. . . . . . 7 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
8	7	csbeq1d 3903	. . . . . 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 2791	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
10	2, 3, 9	csbie2 3938	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵
11	1, 10	eqtr4di 2795	. . 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 9216	. . . 4 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → ^◡𝐴 ∈ Fin)
15		relcnv 6122	. . . . 5 ⊢ Rel ^◡𝐴
16		cnvf1o 8136	. . . . 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 6209	. . . . . 6 ⊢ (Rel 𝐴 ↔ ^◡^◡𝐴 = 𝐴)
20	18, 19	sylib 218	. . . . 5 ⊢ (𝜑 → ^◡^◡𝐴 = 𝐴)
21	20	f1oeq3d 6845	. . . 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 8064	. . . . . . 7 ⊢ ((Rel ^◡𝐴 ∧ 𝑦 ∈ ^◡𝐴) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
24	15, 23	mpan 690	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
25	24	fveq2d 6910	. . . . 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 2842	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ ^◡𝐴)
28		sneq 4636	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → {𝑧} = {⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
29	28	cnveqd 5886	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ^◡{𝑧} = ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
30	29	unieqd 4920	. . . . . . . 8 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ∪ ^◡{𝑧} = ∪ ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
31		opswap 6249	. . . . . . . 8 ⊢ ∪ ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩
32	30, 31	eqtrdi 2793	. . . . . . 7 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ∪ ^◡{𝑧} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
33		eqid 2737	. . . . . . 7 ⊢ (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}) = (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})
34		opex 5469	. . . . . . 7 ⊢ ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ V
35	32, 33, 34	fvmpt 7016	. . . . . 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 2777	. . . 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 15759	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
41		csbeq1a 3913	. . . . 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 5469	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
44		fsumcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
45	43, 44	csbie 3934	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
46		opeq12 4875	. . . . . . . 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 3903	. . . . . 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 2791	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
50	2, 3, 49	csbie2 3938	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶
51	42, 50	eqtr4di 2795	. . 3 ⊢ (𝑦 ∈ ^◡𝐴 → 𝐶 = ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
52	51	sumeq2i 15734	. 2 ⊢ Σ𝑦 ∈ ^◡ 𝐴𝐶 = Σ𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷
53	40, 52	eqtr4di 2795	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ^◡ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⦋csb 3899 {csn 4626 ⟨cop 4632 ∪ cuni 4907 ↦ cmpt 5225 ^◡ccnv 5684 Rel wrel 5690 –1-1-onto→wf1o 6560 ‘cfv 6561 1^st c1st 8012 2^nd c2nd 8013 Fincfn 8985 ℂcc 11153 Σcsu 15722

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723

This theorem is referenced by: fsumcom2 15810

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