Theorem mbfmco2 34890

Description: The pair building of two measurable functions is measurable. ( cf. cnmpt1t 23977). (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 34880	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆)
5	4	ffvelcdmda 7082	. . . . 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 34880	. . . . . 6 ⊢ (𝜑 → 𝐺:∪ 𝑅⟶∪ 𝑇)
9	8	ffvelcdmda 7082	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (𝐺‘𝑥) ∈ ∪ 𝑇)
10		opelxpi 5688	. . . . 5 ⊢ (((𝐹‘𝑥) ∈ ∪ 𝑆 ∧ (𝐺‘𝑥) ∈ ∪ 𝑇) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (∪ 𝑆 × ∪ 𝑇))
11	5, 9, 10	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (∪ 𝑆 × ∪ 𝑇))
12		sxuni 34819	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
13	2, 6, 12	syl2anc 596	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
14	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
15	11, 14	eleqtrd 2863	. . 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 7112	. 2 ⊢ (𝜑 → 𝐻:∪ 𝑅⟶∪ (𝑆 ×s 𝑇))
18		eqid 2761	. . . . 5 ⊢ (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))
19		vex 3455	. . . . . 6 ⊢ 𝑎 ∈ V
20		vex 3455	. . . . . 6 ⊢ 𝑏 ∈ V
21	19, 20	xpex 7765	. . . . 5 ⊢ (𝑎 × 𝑏) ∈ V
22	18, 21	elrnmpo 7554	. . . 4 ⊢ (𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ↔ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏))
23		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑐 = (𝑎 × 𝑏))
24	23	imaeq2d 6052	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) = (◡𝐻 “ (𝑎 × 𝑏)))
25		simp1 1154	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝜑)
26		simp2l 1218	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑎 ∈ 𝑆)
27		simp2r 1219	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑏 ∈ 𝑇)
28	4, 8, 16	xppreima2 33238	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐻 “ (𝑎 × 𝑏)) = ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)))
29	28	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐻 “ (𝑎 × 𝑏)) = ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)))
30	1	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑅 ∈ ∪ ran sigAlgebra)
31	2	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
32	3	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
33		simp2 1155	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑎 ∈ 𝑆)
34	30, 31, 32, 33	mbfmcnvima 34881	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐹 “ 𝑎) ∈ 𝑅)
35	6	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
36	7	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝐺 ∈ (𝑅MblFnM𝑇))
37		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑏 ∈ 𝑇)
38	30, 35, 36, 37	mbfmcnvima 34881	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐺 “ 𝑏) ∈ 𝑅)
39		inelsiga 34761	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ∪ ran sigAlgebra ∧ (◡𝐹 “ 𝑎) ∈ 𝑅 ∧ (◡𝐺 “ 𝑏) ∈ 𝑅) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅)
40	30, 34, 38, 39	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅)
41	29, 40	eqeltrd 2861	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
42	25, 26, 27, 41	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
43	24, 42	eqeltrd 2861	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) ∈ 𝑅)
44	43	3expia 1139	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇)) → (𝑐 = (𝑎 × 𝑏) → (◡𝐻 “ 𝑐) ∈ 𝑅))
45	44	rexlimdvva 3220	. . . . 5 ⊢ (𝜑 → (∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏) → (◡𝐻 “ 𝑐) ∈ 𝑅))
46	45	imp 412	. . . 4 ⊢ ((𝜑 ∧ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) ∈ 𝑅)
47	22, 46	sylan2b 606	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))) → (◡𝐻 “ 𝑐) ∈ 𝑅)
48	47	ralrimiva 3155	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))(◡𝐻 “ 𝑐) ∈ 𝑅)
49		eqid 2761	. . . . 5 ⊢ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))
50	49	txbasex 23878	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
51	2, 6, 50	syl2anc 596	. . 3 ⊢ (𝜑 → ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ∈ V)
52	49	sxval 34816	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))))
53	2, 6, 52	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))))
54	51, 1, 53	imambfm 34887	. 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 726	1 ⊢ (𝜑 → 𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∩ cin 3898  ⟨cop 4590  ∪ cuni 4867   ↦ cmpt 5186   × cxp 5649  ^◡ccnv 5650  ran crn 5652   “ cima 5654  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  sigAlgebracsiga 34733  sigaGencsigagen 34764   ×_s csx 34814  MblFnMcmbfm 34875

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-oi 9497  df-dju 9975  df-card 10013  df-acn 10016  df-siga 34734  df-sigagen 34765  df-sx 34815  df-mbfm 34876

This theorem is used by:  rrvadd  35077