fsumcnv - Metamath Proof Explorer

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Theorem fsumcnv 15861

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 3864	. . . 4 ⊢ (𝑥 = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		fvex 6895	. . . . 5 ⊢ (2^nd ‘𝑦) ∈ V
3		fvex 6895	. . . . 5 ⊢ (1^st ‘𝑦) ∈ V
4		opex 5443	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
5		fsumcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
6	4, 5	csbie 3885	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
7		opeq12 4838	. . . . . . 7 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
8	7	csbeq1d 3854	. . . . . 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 2811	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
10	2, 3, 9	csbie2 3889	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵
11	1, 10	eqtr4di 2815	. . 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 9173	. . . 4 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)
14	12, 13	syl 18	. . 3 ⊢ (𝜑 → ^◡𝐴 ∈ Fin)
15		relcnv 6104	. . . . 5 ⊢ Rel ^◡𝐴
16		cnvf1o 8111	. . . . 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 6186	. . . . . 6 ⊢ (Rel 𝐴 ↔ ^◡^◡𝐴 = 𝐴)
20	18, 19	sylib 221	. . . . 5 ⊢ (𝜑 → ^◡^◡𝐴 = 𝐴)
21	20	f1oeq3d 6818	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}):^◡𝐴–1-1-onto→^◡^◡𝐴 ↔ (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}):^◡𝐴–1-1-onto→𝐴))
22	17, 21	mpbii 236	. . 3 ⊢ (𝜑 → (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}):^◡𝐴–1-1-onto→𝐴)
23		1st2nd 8039	. . . . . . 7 ⊢ ((Rel ^◡𝐴 ∧ 𝑦 ∈ ^◡𝐴) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
24	15, 23	mpan 703	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
25	24	fveq2d 6886	. . . . 5 ⊢ (𝑦 ∈ ^◡𝐴 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘𝑦) = ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩))
26		id 23	. . . . . . 7 ⊢ (𝑦 ∈ ^◡𝐴 → 𝑦 ∈ ^◡𝐴)
27	24, 26	eqeltrrd 2863	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ ^◡𝐴)
28		sneq 4597	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → {𝑧} = {⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
29	28	cnveqd 5859	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ^◡{𝑧} = ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
30	29	unieqd 4883	. . . . . . . 8 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ∪ ^◡{𝑧} = ∪ ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
31		opswap 6229	. . . . . . . 8 ⊢ ∪ ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩
32	30, 31	eqtrdi 2813	. . . . . . 7 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ∪ ^◡{𝑧} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
33		eqid 2762	. . . . . . 7 ⊢ (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}) = (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})
34		opex 5443	. . . . . . 7 ⊢ ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ V
35	32, 33, 34	fvmpt 6990	. . . . . 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 18	. . . . 5 ⊢ (𝑦 ∈ ^◡𝐴 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
37	25, 36	eqtrd 2797	. . . 4 ⊢ (𝑦 ∈ ^◡𝐴 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
38	37	adantl 487	. . 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 15811	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
41		csbeq1a 3864	. . . . 5 ⊢ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → 𝐶 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
42	24, 41	syl 18	. . . 4 ⊢ (𝑦 ∈ ^◡𝐴 → 𝐶 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
43		opex 5443	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
44		fsumcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
45	43, 44	csbie 3885	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
46		opeq12 4838	. . . . . . . 8 ⊢ ((𝑘 = (1^st ‘𝑦) ∧ 𝑗 = (2^nd ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
47	46	ancoms 464	. . . . . . 7 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
48	47	csbeq1d 3854	. . . . . 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 2811	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
50	2, 3, 49	csbie2 3889	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶
51	42, 50	eqtr4di 2815	. . 3 ⊢ (𝑦 ∈ ^◡𝐴 → 𝐶 = ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
52	51	sumeq2i 15787	. 2 ⊢ Σ𝑦 ∈ ^◡ 𝐴𝐶 = Σ𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷
53	40, 52	eqtr4di 2815	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑦 ∈ ^◡ 𝐴𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⦋csb 3850 {csn 4587 ⟨cop 4593 ∪ cuni 4870 ↦ cmpt 5190 ^◡ccnv 5658 Rel wrel 5664 –1-1-onto→wf1o 6536 ‘cfv 6537 1^st c1st 7987 2^nd c2nd 7988 Fincfn 8955 ℂcc 11125 Σcsu 15775

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-z 12619 df-uz 12891 df-rp 13045 df-fz 13564 df-fzo 13712 df-seq 14068 df-exp 14128 df-hash 14397 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-clim 15577 df-sum 15776

This theorem is used by: fsumcom2 15862

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