Theorem muldmmbl 48118

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 4124	. . 3 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
5		eqid 2763	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷))
6		ovexd 7486	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (𝐶 · 𝐷) ∈ V)
7	1, 4, 5, 6	dmmptdff 46509	. 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 47635	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑆)
12	7, 11	eqeltrd 2864	1 ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  Ⅎwnf 1816   ∈ wcel 2147  Ⅎwnfc 2911  Vcvv 3454   ∩ cin 4092   ↦ cmpt 5192  dom cdm 5658  (class class class)co 7451   · cmul 11244  SAlgcsalg 47591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-salg 47592

This theorem is used by: (None)