Theorem muldmmbl 44603

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
muldmmbl.1	⊢ Ⅎ𝑥𝜑
muldmmbl.2	⊢ Ⅎ𝑥𝐴
muldmmbl.3	⊢ Ⅎ𝑥𝐵
muldmmbl.4	⊢ (𝜑 → 𝑆 ∈ SAlg)
muldmmbl.5	⊢ (𝜑 → 𝐴 ∈ 𝑆)
muldmmbl.6	⊢ (𝜑 → 𝐵 ∈ 𝑆)

Assertion

Ref	Expression
muldmmbl	⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) ∈ 𝑆)

Proof of Theorem muldmmbl

Step	Hyp	Ref	Expression
1		muldmmbl.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		muldmmbl.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		muldmmbl.3	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfin 4156	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
5		eqid 2736	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷))
6		ovexd 7342	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (𝐶 · 𝐷) ∈ V)
7	1, 4, 5, 6	dmmptdff 42983	. 2 ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) = (𝐴 ∩ 𝐵))
8		muldmmbl.4	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
9		muldmmbl.5	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
10		muldmmbl.6	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
11	8, 9, 10	salincld 44120	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑆)
12	7, 11	eqeltrd 2837	1 ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 397  Ⅎwnf 1783   ∈ wcel 2104  Ⅎwnfc 2885  Vcvv 3437   ∩ cin 3891   ↦ cmpt 5164  dom cdm 5600  (class class class)co 7307   · cmul 10926  SAlgcsalg 44078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-inf2 9447

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-om 7745  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-er 8529  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-salg 44079

This theorem is referenced by: (None)