Theorem muldmmbl 46456

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 4217	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
5		eqid 2726	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷))
6		ovexd 7459	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (𝐶 · 𝐷) ∈ V)
7	1, 4, 5, 6	dmmptdff 44830	. 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 45973	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑆)
12	7, 11	eqeltrd 2826	1 ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 394  Ⅎwnf 1778   ∈ wcel 2099  Ⅎwnfc 2876  Vcvv 3462   ∩ cin 3946   ↦ cmpt 5236  dom cdm 5682  (class class class)co 7424   · cmul 11163  SAlgcsalg 45929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-salg 45930

This theorem is referenced by: (None)