Theorem cnmptk1 22832

Description: The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)

Hypotheses

Ref	Expression
cnmptk1.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmptk1.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmptk1.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmptk1.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
cnmptk1.b	⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀))
cnmptk1.c	⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cnmptk1	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑀,𝑦   𝑥,𝑧,𝑍,𝑦   𝑧,𝐴   𝑥,𝐵   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑧,𝐶   𝑦,𝐵

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑥,𝑦)   𝐵(𝑧)   𝐶(𝑥,𝑦)   𝐽(𝑧)   𝐾(𝑧)   𝐿(𝑧)   𝑀(𝑧)   𝑋(𝑧)   𝑌(𝑧)

Proof of Theorem cnmptk1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmptk1.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
2	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
3		cnmptk1.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
4	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘𝑍))
5		cnmptk1.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
6		topontop 22062	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
7	1, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
8		topontop 22062	. . . . . . . . . 10 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
9	3, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
10		eqid 2738	. . . . . . . . . 10 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
11	10	xkotopon 22751	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
12	7, 9, 11	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
13		cnmptk1.a	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
14		cnf2 22400	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
15	5, 12, 13, 14	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
16	15	fvmptelrn 6987	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
17		cnf2 22400	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
18	2, 4, 16, 17	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
19		eqid 2738	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
20	19	fmpt 6984	. . . . 5 ⊢ (∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
21	18, 20	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
22		eqidd 2739	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴))
23		eqidd 2739	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵))
24		cnmptk1.c	. . . 4 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
25	21, 22, 23, 24	fmptcof 7002	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑦 ∈ 𝑌 ↦ 𝐶))
26	25	mpteq2dva 5174	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))
27		cnmptk1.b	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀))
28	7, 27	xkoco2cn 22809	. . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ 𝑤)) ∈ ((𝐿 ↑ko 𝐾) Cn (𝑀 ↑ko 𝐾)))
29		coeq2 5767	. . 3 ⊢ (𝑤 = (𝑦 ∈ 𝑌 ↦ 𝐴) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ 𝑤) = ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴)))
30	5, 13, 12, 28, 29	cnmpt11 22814	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑦 ∈ 𝑌 ↦ 𝐴))) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
31	26, 30	eqeltrrd 2840	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ↦ cmpt 5157   ∘ ccom 5593  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Topctop 22042  TopOnctopon 22059   Cn ccn 22375   ↑_ko cxko 22712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-cmp 22538  df-xko 22714

This theorem is referenced by:  cnmpt2k  22839