fprodunsn - Intuitionistic Logic Explorer

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Theorem fprodunsn 12130

Description: Multiply in an additional term in a finite product. See also fprodsplitsn 12159 which is the same but with a Ⅎ𝑘𝜑 hypothesis in place of the distinct variable condition between 𝜑 and 𝑘. (Contributed by Jim Kingdon, 16-Aug-2024.)

**Hypotheses**
Ref	Expression
fprodunsn.f	⊢ Ⅎ𝑘𝐷
fprodunsn.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodunsn.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
fprodunsn.ba	⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)
fprodunsn.ccl	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
fprodunsn.dcl	⊢ (𝜑 → 𝐷 ∈ ℂ)
fprodunsn.d	⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷)

**Assertion**
Ref	Expression
fprodunsn	⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝑉 𝜑,𝑘 Allowed substitution hints: 𝐶(𝑘) 𝐷(𝑘)

Proof of Theorem fprodunsn Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprodunsn.ba	. . . 4 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)
2		disjsn 3728	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
3	1, 2	sylibr 134	. . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅)
4		eqidd 2230	. . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵}))
5		fprodunsn.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		fprodunsn.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		unsnfi 7092	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin)
8	5, 6, 1, 7	syl3anc 1271	. . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin)
9		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴)
10	9	orcd 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
11		df-dc 840	. . . . . 6 ⊢ (DECID 𝑗 ∈ 𝐴 ↔ (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
12	10, 11	sylibr 134	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → DECID 𝑗 ∈ 𝐴)
13		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → 𝑗 ∈ {𝐵})
14		velsn 3683	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝐵} ↔ 𝑗 = 𝐵)
15	13, 14	sylib 122	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → 𝑗 = 𝐵)
16	1	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → ¬ 𝐵 ∈ 𝐴)
17	15, 16	eqneltrd 2325	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → ¬ 𝑗 ∈ 𝐴)
18	17	olcd 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
19	18, 11	sylibr 134	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → DECID 𝑗 ∈ 𝐴)
20		elun 3345	. . . . . . 7 ⊢ (𝑗 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑗 ∈ 𝐴 ∨ 𝑗 ∈ {𝐵}))
21	20	biimpi 120	. . . . . 6 ⊢ (𝑗 ∈ (𝐴 ∪ {𝐵}) → (𝑗 ∈ 𝐴 ∨ 𝑗 ∈ {𝐵}))
22	21	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) → (𝑗 ∈ 𝐴 ∨ 𝑗 ∈ {𝐵}))
23	12, 19, 22	mpjaodan 803	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) → DECID 𝑗 ∈ 𝐴)
24	23	ralrimiva 2603	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ (𝐴 ∪ {𝐵})DECID 𝑗 ∈ 𝐴)
25		fprodunsn.ccl	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
26	25	adantlr 477	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
27		elsni 3684	. . . . . . 7 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵)
28	27	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝑘 = 𝐵)
29		fprodunsn.d	. . . . . 6 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷)
30	28, 29	syl 14	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝐶 = 𝐷)
31		fprodunsn.dcl	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ)
32	31	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝐷 ∈ ℂ)
33	30, 32	eqeltrd 2306	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝐶 ∈ ℂ)
34		elun 3345	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}))
35	34	biimpi 120	. . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}))
36	35	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}))
37	26, 33, 36	mpjaodan 803	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ)
38	3, 4, 8, 24, 37	fprodsplitdc 12122	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶))
39		fprodunsn.f	. . . . 5 ⊢ Ⅎ𝑘𝐷
40	39, 29	prodsnf 12118	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷)
41	6, 31, 40	syl2anc 411	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷)
42	41	oveq2d 6023	. 2 ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶) = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))
43	38, 42	eqtrd 2262	1 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 713 DECID wdc 839 = wceq 1395 ∈ wcel 2200 Ⅎwnfc 2359 ∪ cun 3195 ∩ cin 3196 ∅c0 3491 {csn 3666 (class class class)co 6007 Fincfn 6895 ℂcc 8008 · cmul 8015 ∏cprod 12076

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-oadd 6572 df-er 6688 df-en 6896 df-dom 6897 df-fin 6898 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-fz 10217 df-fzo 10351 df-seqfrec 10682 df-exp 10773 df-ihash 11010 df-cj 11368 df-re 11369 df-im 11370 df-rsqrt 11524 df-abs 11525 df-clim 11805 df-proddc 12077

This theorem is referenced by: fprodcl2lem 12131 fprodconst 12146 fprodap0 12147 fprodrec 12155 fprodmodd 12167

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