Theorem fprodcnv 11504

Description: Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)

Hypotheses

Ref	Expression
fprodcnv.1	⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
fprodcnv.2	⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
fprodcnv.3	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodcnv.4	⊢ (𝜑 → Rel 𝐴)
fprodcnv.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
fprodcnv	⊢ (𝜑 → ∏𝑥 ∈ 𝐴 𝐵 = ∏𝑦 ∈ ◡ 𝐴𝐶)

Distinct variable groups:   𝑥,𝐴,𝑦   𝐵,𝑗,𝑘,𝑦   𝐶,𝑗,𝑘   𝑥,𝐷,𝑦   𝑗,𝑘,𝑥,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑗,𝑘)   𝐴(𝑗,𝑘)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑗,𝑘)

Proof of Theorem fprodcnv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1a 3040	. . . 4 ⊢ (𝑥 = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		2ndexg 6110	. . . . . 6 ⊢ (𝑦 ∈ V → (2nd ‘𝑦) ∈ V)
3	2	elv 2716	. . . . 5 ⊢ (2nd ‘𝑦) ∈ V
4		1stexg 6109	. . . . . 6 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
5	4	elv 2716	. . . . 5 ⊢ (1st ‘𝑦) ∈ V
6		vex 2715	. . . . . . . 8 ⊢ 𝑗 ∈ V
7		vex 2715	. . . . . . . 8 ⊢ 𝑘 ∈ V
8	6, 7	opex 4188	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
9		fprodcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
10	8, 9	csbie 3076	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
11		opeq12 3743	. . . . . . 7 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
12	11	csbeq1d 3038	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
13	10, 12	syl5eqr 2204	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵)
14	3, 5, 13	csbie2 3080	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ / 𝑥⦌𝐵
15	1, 14	eqtr4di 2208	. . 3 ⊢ (𝑥 = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ → 𝐵 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
16		fprodcnv.4	. . . 4 ⊢ (𝜑 → Rel 𝐴)
17		fprodcnv.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
18		relcnvfi 6878	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin)
19	16, 17, 18	syl2anc 409	. . 3 ⊢ (𝜑 → ◡𝐴 ∈ Fin)
20		relcnv 4961	. . . . 5 ⊢ Rel ◡𝐴
21		cnvf1o 6166	. . . . 5 ⊢ (Rel ◡𝐴 → (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴)
22	20, 21	ax-mp 5	. . . 4 ⊢ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴
23		dfrel2 5033	. . . . . 6 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
24	16, 23	sylib 121	. . . . 5 ⊢ (𝜑 → ◡◡𝐴 = 𝐴)
25	24	f1oeq3d 5408	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→◡◡𝐴 ↔ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→𝐴))
26	22, 25	mpbii 147	. . 3 ⊢ (𝜑 → (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}):◡𝐴–1-1-onto→𝐴)
27		1st2nd 6123	. . . . . . 7 ⊢ ((Rel ◡𝐴 ∧ 𝑦 ∈ ◡𝐴) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
28	20, 27	mpan 421	. . . . . 6 ⊢ (𝑦 ∈ ◡𝐴 → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
29	28	fveq2d 5469	. . . . 5 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩))
30	28	eleq1d 2226	. . . . . . 7 ⊢ (𝑦 ∈ ◡𝐴 → (𝑦 ∈ ◡𝐴 ↔ ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ◡𝐴))
31	30	ibi 175	. . . . . 6 ⊢ (𝑦 ∈ ◡𝐴 → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ◡𝐴)
32		sneq 3571	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → {𝑧} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
33	32	cnveqd 4759	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ◡{𝑧} = ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
34	33	unieqd 3783	. . . . . . . 8 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
35		opswapg 5069	. . . . . . . . 9 ⊢ (((1st ‘𝑦) ∈ V ∧ (2nd ‘𝑦) ∈ V) → ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
36	5, 3, 35	mp2an 423	. . . . . . . 8 ⊢ ∪ ◡{⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩
37	34, 36	eqtrdi 2206	. . . . . . 7 ⊢ (𝑧 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → ∪ ◡{𝑧} = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
38		eqid 2157	. . . . . . 7 ⊢ (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧}) = (𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})
39	3, 5	opex 4188	. . . . . . 7 ⊢ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩ ∈ V
40	37, 38, 39	fvmpt 5542	. . . . . 6 ⊢ (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
41	31, 40	syl 14	. . . . 5 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
42	29, 41	eqtrd 2190	. . . 4 ⊢ (𝑦 ∈ ◡𝐴 → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
43	42	adantl 275	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ◡𝐴) → ((𝑧 ∈ ◡𝐴 ↦ ∪ ◡{𝑧})‘𝑦) = ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
44		fprodcnv.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
45	15, 19, 26, 43, 44	fprodf1o 11467	. 2 ⊢ (𝜑 → ∏𝑥 ∈ 𝐴 𝐵 = ∏𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
46		csbeq1a 3040	. . . . 5 ⊢ (𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ → 𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
47	28, 46	syl 14	. . . 4 ⊢ (𝑦 ∈ ◡𝐴 → 𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
48	7, 6	opex 4188	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
49		fprodcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
50	48, 49	csbie 3076	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
51		opeq12 3743	. . . . . . . 8 ⊢ ((𝑘 = (1st ‘𝑦) ∧ 𝑗 = (2nd ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
52	51	ancoms 266	. . . . . . 7 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
53	52	csbeq1d 3038	. . . . . 6 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
54	50, 53	syl5eqr 2204	. . . . 5 ⊢ ((𝑗 = (2nd ‘𝑦) ∧ 𝑘 = (1st ‘𝑦)) → 𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶)
55	3, 5, 54	csbie2 3080	. . . 4 ⊢ ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ / 𝑦⦌𝐶
56	47, 55	eqtr4di 2208	. . 3 ⊢ (𝑦 ∈ ◡𝐴 → 𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷)
57	56	prodeq2i 11441	. 2 ⊢ ∏𝑦 ∈ ◡ 𝐴𝐶 = ∏𝑦 ∈ ◡ 𝐴⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑘⦌𝐷
58	45, 57	eqtr4di 2208	1 ⊢ (𝜑 → ∏𝑥 ∈ 𝐴 𝐵 = ∏𝑦 ∈ ◡ 𝐴𝐶)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∈ wcel 2128  Vcvv 2712  ⦋csb 3031  {csn 3560  ⟨cop 3563  ∪ cuni 3772   ↦ cmpt 4025  ^◡ccnv 4582  Rel wrel 4588  –1-1-onto→wf1o 5166  ‘cfv 5167  1^st c1st 6080  2^nd c2nd 6081  Fincfn 6678  ℂcc 7713  ∏cprod 11429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-frec 6332  df-1o 6357  df-oadd 6361  df-er 6473  df-en 6679  df-dom 6680  df-fin 6681  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-fz 9895  df-fzo 10024  df-seqfrec 10327  df-exp 10401  df-ihash 10632  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-clim 11158  df-proddc 11430

This theorem is referenced by:  fprodcom2fi  11505