fprodunsn - Intuitionistic Logic Explorer

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Theorem fprodunsn 11614

Description: Multiply in an additional term in a finite product. See also fprodsplitsn 11643 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 3656	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
3	1, 2	sylibr 134	. . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅)
4		eqidd 2178	. . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵}))
5		fprodunsn.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		fprodunsn.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		unsnfi 6920	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin)
8	5, 6, 1, 7	syl3anc 1238	. . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin)
9		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴)
10	9	orcd 733	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
11		df-dc 835	. . . . . 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 3611	. . . . . . . . 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 2273	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → ¬ 𝑗 ∈ 𝐴)
18	17	olcd 734	. . . . . 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 3278	. . . . . . 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 798	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) → DECID 𝑗 ∈ 𝐴)
24	23	ralrimiva 2550	. . 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 3612	. . . . . . 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 2254	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝐶 ∈ ℂ)
34		elun 3278	. . . . . 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 798	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ)
38	3, 4, 8, 24, 37	fprodsplitdc 11606	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶))
39		fprodunsn.f	. . . . 5 ⊢ Ⅎ𝑘𝐷
40	39, 29	prodsnf 11602	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷)
41	6, 31, 40	syl2anc 411	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷)
42	41	oveq2d 5893	. 2 ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶) = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))
43	38, 42	eqtrd 2210	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 708 DECID wdc 834 = wceq 1353 ∈ wcel 2148 Ⅎwnfc 2306 ∪ cun 3129 ∩ cin 3130 ∅c0 3424 {csn 3594 (class class class)co 5877 Fincfn 6742 ℂcc 7811 · cmul 7818 ∏cprod 11560

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-en 6743 df-dom 6744 df-fin 6745 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-ihash 10758 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-proddc 11561

This theorem is referenced by: fprodcl2lem 11615 fprodconst 11630 fprodap0 11631 fprodrec 11639 fprodmodd 11651

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