fprodcnv - Metamath Proof Explorer

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Theorem fprodcnv 15927

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 3908	. . . 4 ⊢ (𝑥 = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ → 𝐵 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
2		fvex 6905	. . . . 5 ⊢ (2^nd ‘𝑦) ∈ V
3		fvex 6905	. . . . 5 ⊢ (1^st ‘𝑦) ∈ V
4		opex 5465	. . . . . . 7 ⊢ ⟨𝑗, 𝑘⟩ ∈ V
5		fprodcnv.1	. . . . . . 7 ⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
6	4, 5	csbie 3930	. . . . . 6 ⊢ ⦋⟨𝑗, 𝑘⟩ / 𝑥⦌𝐵 = 𝐷
7		opeq12 4876	. . . . . . 7 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
8	7	csbeq1d 3898	. . . . . 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 2787	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵)
10	2, 3, 9	csbie2 3936	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ / 𝑥⦌𝐵
11	1, 10	eqtr4di 2791	. . 3 ⊢ (𝑥 = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ → 𝐵 = ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
12		fprodcnv.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
13		cnvfi 9180	. . . 4 ⊢ (𝐴 ∈ Fin → ^◡𝐴 ∈ Fin)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → ^◡𝐴 ∈ Fin)
15		relcnv 6104	. . . . 5 ⊢ Rel ^◡𝐴
16		cnvf1o 8097	. . . . 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		fprodcnv.4	. . . . . 6 ⊢ (𝜑 → Rel 𝐴)
19		dfrel2 6189	. . . . . 6 ⊢ (Rel 𝐴 ↔ ^◡^◡𝐴 = 𝐴)
20	18, 19	sylib 217	. . . . 5 ⊢ (𝜑 → ^◡^◡𝐴 = 𝐴)
21	20	f1oeq3d 6831	. . . 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 8025	. . . . . . 7 ⊢ ((Rel ^◡𝐴 ∧ 𝑦 ∈ ^◡𝐴) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
24	15, 23	mpan 689	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
25	24	fveq2d 6896	. . . . 5 ⊢ (𝑦 ∈ ^◡𝐴 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘𝑦) = ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩))
26	24	eleq1d 2819	. . . . . . 7 ⊢ (𝑦 ∈ ^◡𝐴 → (𝑦 ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ ^◡𝐴))
27	26	ibi 267	. . . . . 6 ⊢ (𝑦 ∈ ^◡𝐴 → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ ^◡𝐴)
28		sneq 4639	. . . . . . . . . 10 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → {𝑧} = {⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
29	28	cnveqd 5876	. . . . . . . . 9 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ^◡{𝑧} = ^◡{⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩})
30	29	unieqd 4923	. . . . . . . 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 2789	. . . . . . 7 ⊢ (𝑧 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → ∪ ^◡{𝑧} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
33		eqid 2733	. . . . . . 7 ⊢ (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧}) = (𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})
34		opex 5465	. . . . . . 7 ⊢ ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ V
35	32, 33, 34	fvmpt 6999	. . . . . 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 2773	. . . 4 ⊢ (𝑦 ∈ ^◡𝐴 → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
38	37	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ^◡𝐴) → ((𝑧 ∈ ^◡𝐴 ↦ ∪ ^◡{𝑧})‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
39		fprodcnv.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
40	11, 14, 22, 38, 39	fprodf1o 15890	. 2 ⊢ (𝜑 → ∏𝑥 ∈ 𝐴 𝐵 = ∏𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
41		csbeq1a 3908	. . . . 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 5465	. . . . . . 7 ⊢ ⟨𝑘, 𝑗⟩ ∈ V
44		fprodcnv.2	. . . . . . 7 ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
45	43, 44	csbie 3930	. . . . . 6 ⊢ ⦋⟨𝑘, 𝑗⟩ / 𝑦⦌𝐶 = 𝐷
46		opeq12 4876	. . . . . . . 8 ⊢ ((𝑘 = (1^st ‘𝑦) ∧ 𝑗 = (2^nd ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
47	46	ancoms 460	. . . . . . 7 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
48	47	csbeq1d 3898	. . . . . 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 2787	. . . . 5 ⊢ ((𝑗 = (2^nd ‘𝑦) ∧ 𝑘 = (1^st ‘𝑦)) → 𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶)
50	2, 3, 49	csbie2 3936	. . . 4 ⊢ ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷 = ⦋⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ / 𝑦⦌𝐶
51	42, 50	eqtr4di 2791	. . 3 ⊢ (𝑦 ∈ ^◡𝐴 → 𝐶 = ⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷)
52	51	prodeq2i 15863	. 2 ⊢ ∏𝑦 ∈ ^◡ 𝐴𝐶 = ∏𝑦 ∈ ^◡ 𝐴⦋(2^nd ‘𝑦) / 𝑗⦌⦋(1^st ‘𝑦) / 𝑘⦌𝐷
53	40, 52	eqtr4di 2791	1 ⊢ (𝜑 → ∏𝑥 ∈ 𝐴 𝐵 = ∏𝑦 ∈ ^◡ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⦋csb 3894 {csn 4629 ⟨cop 4635 ∪ cuni 4909 ↦ cmpt 5232 ^◡ccnv 5676 Rel wrel 5682 –1-1-onto→wf1o 6543 ‘cfv 6544 1^st c1st 7973 2^nd c2nd 7974 Fincfn 8939 ℂcc 11108 ∏cprod 15849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-prod 15850

This theorem is referenced by: fprodcom2 15928

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