Theorem muldmmbl2 46882

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 5890	. . . . 5 ⊢ Ⅎ𝑥dom 𝐹
3		muldmmbl2.2	. . . . . 6 ⊢ Ⅎ𝑥𝐺
4	3	nfdm 5890	. . . . 5 ⊢ Ⅎ𝑥dom 𝐺
5	2, 4	nfin 4171	. . . 4 ⊢ Ⅎ𝑥(dom 𝐹 ∩ dom 𝐺)
6		ovex 7379	. . . 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 45311	. . 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 46398	. 2 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) ∈ 𝑆)
14	9, 13	eqeltrd 2831	1 ⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Ⅎwnfc 2879   ∩ cin 3896   ↦ cmpt 5170  dom cdm 5614  ‘cfv 6481  (class class class)co 7346   · cmul 11011  SAlgcsalg 46354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-salg 46355

This theorem is referenced by: (None)