Theorem muldmmbl 47079

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 4176	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
5		eqid 2736	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷))
6		ovexd 7393	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (𝐶 · 𝐷) ∈ V)
7	1, 4, 5, 6	dmmptdff 45467	. 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 46596	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑆)
12	7, 11	eqeltrd 2836	1 ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2113  Ⅎwnfc 2883  Vcvv 3440   ∩ cin 3900   ↦ cmpt 5179  dom cdm 5624  (class class class)co 7358   · cmul 11031  SAlgcsalg 46552

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-salg 46553

This theorem is referenced by: (None)