Theorem mbfmco2 35008

Description: The pair building of two measurable functions is measurable. ( cf. cnmpt1t 24039). (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 34998	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆)
5	4	ffvelcdmda 7116	. . . . 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 34998	. . . . . 6 ⊢ (𝜑 → 𝐺:∪ 𝑅⟶∪ 𝑇)
9	8	ffvelcdmda 7116	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (𝐺‘𝑥) ∈ ∪ 𝑇)
10		opelxpi 5695	. . . . 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 34937	. . . . . 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 2866	. . 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 7146	. 2 ⊢ (𝜑 → 𝐻:∪ 𝑅⟶∪ (𝑆 ×s 𝑇))
18		eqid 2763	. . . . 5 ⊢ (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))
19		vex 3458	. . . . . 6 ⊢ 𝑎 ∈ V
20		vex 3458	. . . . . 6 ⊢ 𝑏 ∈ V
21	19, 20	xpex 7799	. . . . 5 ⊢ (𝑎 × 𝑏) ∈ V
22	18, 21	elrnmpo 7588	. . . 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 6070	. . . . . . . 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 33343	. . . . . . . . . . 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 34999	. . . . . . . . . . 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 34999	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐺 “ 𝑏) ∈ 𝑅)
39		inelsiga 34879	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ∪ ran sigAlgebra ∧ (◡𝐹 “ 𝑎) ∈ 𝑅 ∧ (◡𝐺 “ 𝑏) ∈ 𝑅) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅)
40	30, 34, 38, 39	syl3anc 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅)
41	29, 40	eqeltrd 2864	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
42	25, 26, 27, 41	syl3anc 1399	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅)
43	24, 42	eqeltrd 2864	. . . . . . 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 3223	. . . . 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 3158	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))(◡𝐻 “ 𝑐) ∈ 𝑅)
49		eqid 2763	. . . . 5 ⊢ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))
50	49	txbasex 23940	. . . 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 34934	. . . 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 35005	. 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 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ∩ cin 4092  ⟨cop 4840  ∪ cuni 4872   ↦ cmpt 5192   × cxp 5656  ^◡ccnv 5657  ran crn 5659   “ cima 5661  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  sigAlgebracsiga 34851  sigaGencsigagen 34882   ×_s csx 34932  MblFnMcmbfm 34993

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-rep 5238  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-rmo 3369  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-se 5612  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-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  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-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-oi 9541  df-dju 10019  df-card 10057  df-acn 10060  df-siga 34852  df-sigagen 34883  df-sx 34933  df-mbfm 34994

This theorem is used by:  rrvadd  35195