Theorem mbfmco2 34247

Description: The pair building of two measurable functions is measurable. ( cf. cnmpt1t 23689). (Contributed by Thierry Arnoux, 6-Jun-2017.)

Hypotheses

Ref	Expression
mbfmco.1	⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra)
mbfmco.2	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
mbfmco.3	⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
mbfmco2.4	⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆))
mbfmco2.5	⊢ (𝜑 → 𝐺 ∈ (𝑅MblFnM𝑇))
mbfmco2.6	⊢ 𝐻 = (𝑥 ∈ ∪ 𝑅 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)

Assertion

Ref	Expression
mbfmco2	⊢ (𝜑 → 𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))

Distinct variable groups:   𝑥,𝑅   𝑥,𝑆   𝑥,𝑇   𝜑,𝑥   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻

Proof of Theorem mbfmco2

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmco.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra)
2		mbfmco.2	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
3		mbfmco2.4	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆))
4	1, 2, 3	mbfmf 34235	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆)
5	4	ffvelcdmda 7104	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (𝐹‘𝑥) ∈ ∪ 𝑆)
6		mbfmco.3	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
7		mbfmco2.5	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝑅MblFnM𝑇))
8	1, 6, 7	mbfmf 34235	. . . . . 6 ⊢ (𝜑 → 𝐺:∪ 𝑅⟶∪ 𝑇)
9	8	ffvelcdmda 7104	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (𝐺‘𝑥) ∈ ∪ 𝑇)
10		opelxpi 5726	. . . . 5 ⊢ (((𝐹‘𝑥) ∈ ∪ 𝑆 ∧ (𝐺‘𝑥) ∈ ∪ 𝑇) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (∪ 𝑆 × ∪ 𝑇))
11	5, 9, 10	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (∪ 𝑆 × ∪ 𝑇))
12		sxuni 34174	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
13	2, 6, 12	syl2anc 584	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
15	11, 14	eleqtrd 2841	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ ∪ (𝑆 ×s 𝑇))
16		mbfmco2.6	. . 3 ⊢ 𝐻 = (𝑥 ∈ ∪ 𝑅 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
17	15, 16	fmptd 7134	. 2 ⊢ (𝜑 → 𝐻:∪ 𝑅⟶∪ (𝑆 ×s 𝑇))
18		eqid 2735	. . . . 5 ⊢ (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))
19		vex 3482	. . . . . 6 ⊢ 𝑎 ∈ V
20		vex 3482	. . . . . 6 ⊢ 𝑏 ∈ V
21	19, 20	xpex 7772	. . . . 5 ⊢ (𝑎 × 𝑏) ∈ V
22	18, 21	elrnmpo 7569	. . . 4 ⊢ (𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ↔ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏))
23		simp3 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑐 = (𝑎 × 𝑏))
24	23	imaeq2d 6080	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) = (◡𝐻 “ (𝑎 × 𝑏)))
25		simp1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝜑)
26		simp2l 1198	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑎 ∈ 𝑆)
27		simp2r 1199	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑏 ∈ 𝑇)
28	4, 8, 16	xppreima2 32668	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐻 “ (𝑎 × 𝑏)) = ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)))
29	28	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐻 “ (𝑎 × 𝑏)) = ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)))
30	1	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑅 ∈ ∪ ran sigAlgebra)
31	2	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
32	3	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
33		simp2 1136	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑎 ∈ 𝑆)
34	30, 31, 32, 33	mbfmcnvima 34237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐹 “ 𝑎) ∈ 𝑅)
35	6	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
36	7	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝐺 ∈ (𝑅MblFnM𝑇))
37		simp3 1137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑏 ∈ 𝑇)
38	30, 35, 36, 37	mbfmcnvima 34237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐺 “ 𝑏) ∈ 𝑅)
39		inelsiga 34116	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ∪ ran sigAlgebra ∧ (◡𝐹 “ 𝑎) ∈ 𝑅 ∧ (◡𝐺 “ 𝑏) ∈ 𝑅) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅)
40	30, 34, 38, 39	syl3anc 1370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅)
41	29, 40	eqeltrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
42	25, 26, 27, 41	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
43	24, 42	eqeltrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) ∈ 𝑅)
44	43	3expia 1120	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇)) → (𝑐 = (𝑎 × 𝑏) → (◡𝐻 “ 𝑐) ∈ 𝑅))
45	44	rexlimdvva 3211	. . . . 5 ⊢ (𝜑 → (∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏) → (◡𝐻 “ 𝑐) ∈ 𝑅))
46	45	imp 406	. . . 4 ⊢ ((𝜑 ∧ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) ∈ 𝑅)
47	22, 46	sylan2b 594	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))) → (◡𝐻 “ 𝑐) ∈ 𝑅)
48	47	ralrimiva 3144	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))(◡𝐻 “ 𝑐) ∈ 𝑅)
49		eqid 2735	. . . . 5 ⊢ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))
50	49	txbasex 23590	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
51	2, 6, 50	syl2anc 584	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
52	49	sxval 34171	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))))
53	2, 6, 52	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))))
54	51, 1, 53	imambfm 34244	. 2 ⊢ (𝜑 → (𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)) ↔ (𝐻:∪ 𝑅⟶∪ (𝑆 ×s 𝑇) ∧ ∀𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))(◡𝐻 “ 𝑐) ∈ 𝑅)))
55	17, 48, 54	mpbir2and 713	1 ⊢ (𝜑 → 𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ∩ cin 3962  ⟨cop 4637  ∪ cuni 4912   ↦ cmpt 5231   × cxp 5687  ^◡ccnv 5688  ran crn 5690   “ cima 5692  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  sigAlgebracsiga 34089  sigaGencsigagen 34119   ×_s csx 34169  MblFnMcmbfm 34230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-siga 34090  df-sigagen 34120  df-sx 34170  df-mbfm 34231

This theorem is referenced by:  rrvadd  34434