Theorem muldmmbl2 45487

Description: If two functions have domains in the sigma-algebra, the domain of their multiplication also belongs to the sigma-algebra. This is the second statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their multiplication. (Contributed by Glauco Siliprandi, 30-Dec-2024.)

Hypotheses

Ref	Expression
muldmmbl2.1	⊢ Ⅎ𝑥𝐹
muldmmbl2.2	⊢ Ⅎ𝑥𝐺
muldmmbl2.3	⊢ (𝜑 → 𝑆 ∈ SAlg)
muldmmbl2.4	⊢ (𝜑 → dom 𝐹 ∈ 𝑆)
muldmmbl2.5	⊢ (𝜑 → dom 𝐺 ∈ 𝑆)
muldmmbl2.6	⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))

Assertion

Ref	Expression
muldmmbl2	⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

Proof of Theorem muldmmbl2

Step	Hyp	Ref	Expression
1		muldmmbl2.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
2	1	nfdm 5948	. . . . 5 ⊢ Ⅎ𝑥dom 𝐹
3		muldmmbl2.2	. . . . . 6 ⊢ Ⅎ𝑥𝐺
4	3	nfdm 5948	. . . . 5 ⊢ Ⅎ𝑥dom 𝐺
5	2, 4	nfin 4215	. . . 4 ⊢ Ⅎ𝑥(dom 𝐹 ∩ dom 𝐺)
6		ovex 7437	. . . 4 ⊢ ((𝐹‘𝑥) · (𝐺‘𝑥)) ∈ V
7		muldmmbl2.6	. . . 4 ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))
8	5, 6, 7	dmmptif 43906	. . 3 ⊢ dom 𝐻 = (dom 𝐹 ∩ dom 𝐺)
9	8	a1i 11	. 2 ⊢ (𝜑 → dom 𝐻 = (dom 𝐹 ∩ dom 𝐺))
10		muldmmbl2.3	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11		muldmmbl2.4	. . 3 ⊢ (𝜑 → dom 𝐹 ∈ 𝑆)
12		muldmmbl2.5	. . 3 ⊢ (𝜑 → dom 𝐺 ∈ 𝑆)
13	10, 11, 12	salincld 45003	. 2 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) ∈ 𝑆)
14	9, 13	eqeltrd 2834	1 ⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Ⅎwnfc 2884   ∩ cin 3946   ↦ cmpt 5230  dom cdm 5675  ‘cfv 6540  (class class class)co 7404   · cmul 11111  SAlgcsalg 44959

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-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632

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-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-salg 44960

This theorem is referenced by: (None)