Theorem mbfmco2 30327

Description: The pair building of two measurable functions is measurable. ( cf. cnmpt1t 21468). (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 30317	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆)
5	4	ffvelrnda 6359	. . . . 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 30317	. . . . . 6 ⊢ (𝜑 → 𝐺:∪ 𝑅⟶∪ 𝑇)
9	8	ffvelrnda 6359	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (𝐺‘𝑥) ∈ ∪ 𝑇)
10		opelxpi 5148	. . . . 5 ⊢ (((𝐹‘𝑥) ∈ ∪ 𝑆 ∧ (𝐺‘𝑥) ∈ ∪ 𝑇) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (∪ 𝑆 × ∪ 𝑇))
11	5, 9, 10	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (∪ 𝑆 × ∪ 𝑇))
12		sxuni 30256	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
13	2, 6, 12	syl2anc 693	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
14	13	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
15	11, 14	eleqtrd 2703	. . 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 6385	. 2 ⊢ (𝜑 → 𝐻:∪ 𝑅⟶∪ (𝑆 ×s 𝑇))
18		eqid 2622	. . . . 5 ⊢ (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))
19		vex 3203	. . . . . 6 ⊢ 𝑎 ∈ V
20		vex 3203	. . . . . 6 ⊢ 𝑏 ∈ V
21	19, 20	xpex 6962	. . . . 5 ⊢ (𝑎 × 𝑏) ∈ V
22	18, 21	elrnmpt2 6773	. . . 4 ⊢ (𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ↔ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏))
23		simp3 1063	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑐 = (𝑎 × 𝑏))
24	23	imaeq2d 5466	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) = (◡𝐻 “ (𝑎 × 𝑏)))
25		simp1 1061	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝜑)
26		simp2l 1087	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑎 ∈ 𝑆)
27		simp2r 1088	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑏 ∈ 𝑇)
28	4, 8, 16	xppreima2 29450	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐻 “ (𝑎 × 𝑏)) = ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)))
29	28	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐻 “ (𝑎 × 𝑏)) = ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)))
30	1	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑅 ∈ ∪ ran sigAlgebra)
31	2	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
32	3	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
33		simp2 1062	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑎 ∈ 𝑆)
34	30, 31, 32, 33	mbfmcnvima 30319	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐹 “ 𝑎) ∈ 𝑅)
35	6	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
36	7	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝐺 ∈ (𝑅MblFnM𝑇))
37		simp3 1063	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑏 ∈ 𝑇)
38	30, 35, 36, 37	mbfmcnvima 30319	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐺 “ 𝑏) ∈ 𝑅)
39		inelsiga 30198	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ∪ ran sigAlgebra ∧ (◡𝐹 “ 𝑎) ∈ 𝑅 ∧ (◡𝐺 “ 𝑏) ∈ 𝑅) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅)
40	30, 34, 38, 39	syl3anc 1326	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅)
41	29, 40	eqeltrd 2701	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
42	25, 26, 27, 41	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
43	24, 42	eqeltrd 2701	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) ∈ 𝑅)
44	43	3expia 1267	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇)) → (𝑐 = (𝑎 × 𝑏) → (◡𝐻 “ 𝑐) ∈ 𝑅))
45	44	rexlimdvva 3038	. . . . 5 ⊢ (𝜑 → (∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏) → (◡𝐻 “ 𝑐) ∈ 𝑅))
46	45	imp 445	. . . 4 ⊢ ((𝜑 ∧ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) ∈ 𝑅)
47	22, 46	sylan2b 492	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))) → (◡𝐻 “ 𝑐) ∈ 𝑅)
48	47	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))(◡𝐻 “ 𝑐) ∈ 𝑅)
49		eqid 2622	. . . . 5 ⊢ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))
50	49	txbasex 21369	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
51	2, 6, 50	syl2anc 693	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
52	49	sxval 30253	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))))
53	2, 6, 52	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))))
54	51, 1, 53	imambfm 30324	. 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 957	1 ⊢ (𝜑 → 𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∩ cin 3573  ⟨cop 4183  ∪ cuni 4436   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  sigAlgebracsiga 30170  sigaGencsigagen 30201   ×_s csx 30251  MblFnMcmbfm 30312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-siga 30171  df-sigagen 30202  df-sx 30252  df-mbfm 30313

This theorem is referenced by:  rrvadd  30514