ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ctiunctlemudc GIF version

Theorem ctiunctlemudc 11845
Description: Lemma for ctiunct 11848. (Contributed by Jim Kingdon, 28-Oct-2023.)
Hypotheses
Ref Expression
ctiunct.som (𝜑𝑆 ⊆ ω)
ctiunct.sdc (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
ctiunct.f (𝜑𝐹:𝑆onto𝐴)
ctiunct.tom ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)
ctiunct.tdc ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)
ctiunct.g ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)
ctiunct.j (𝜑𝐽:ω–1-1-onto→(ω × ω))
ctiunct.u 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}
Assertion
Ref Expression
ctiunctlemudc (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑈)
Distinct variable groups:   𝑥,𝐴   𝑛,𝐹,𝑥   𝑧,𝐹,𝑥   𝑛,𝐽,𝑥   𝑧,𝐽   𝑆,𝑛   𝑧,𝑆   𝑇,𝑛   𝑧,𝑇   𝑈,𝑛   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑧,𝑛)   𝐴(𝑧,𝑛)   𝐵(𝑥,𝑧,𝑛)   𝑆(𝑥)   𝑇(𝑥)   𝑈(𝑥,𝑧)   𝐺(𝑥,𝑧,𝑛)

Proof of Theorem ctiunctlemudc
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2178 . . . . . . . . 9 (𝑛 = (1st ‘(𝐽𝑚)) → (𝑛𝑆 ↔ (1st ‘(𝐽𝑚)) ∈ 𝑆))
21dcbid 806 . . . . . . . 8 (𝑛 = (1st ‘(𝐽𝑚)) → (DECID 𝑛𝑆DECID (1st ‘(𝐽𝑚)) ∈ 𝑆))
3 ctiunct.sdc . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
43adantr 272 . . . . . . . 8 ((𝜑𝑚 ∈ ω) → ∀𝑛 ∈ ω DECID 𝑛𝑆)
5 ctiunct.j . . . . . . . . . . . 12 (𝜑𝐽:ω–1-1-onto→(ω × ω))
65adantr 272 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ω) → 𝐽:ω–1-1-onto→(ω × ω))
7 f1of 5333 . . . . . . . . . . 11 (𝐽:ω–1-1-onto→(ω × ω) → 𝐽:ω⟶(ω × ω))
86, 7syl 14 . . . . . . . . . 10 ((𝜑𝑚 ∈ ω) → 𝐽:ω⟶(ω × ω))
9 simpr 109 . . . . . . . . . 10 ((𝜑𝑚 ∈ ω) → 𝑚 ∈ ω)
108, 9ffvelrnd 5522 . . . . . . . . 9 ((𝜑𝑚 ∈ ω) → (𝐽𝑚) ∈ (ω × ω))
11 xp1st 6029 . . . . . . . . 9 ((𝐽𝑚) ∈ (ω × ω) → (1st ‘(𝐽𝑚)) ∈ ω)
1210, 11syl 14 . . . . . . . 8 ((𝜑𝑚 ∈ ω) → (1st ‘(𝐽𝑚)) ∈ ω)
132, 4, 12rspcdva 2766 . . . . . . 7 ((𝜑𝑚 ∈ ω) → DECID (1st ‘(𝐽𝑚)) ∈ 𝑆)
1413adantr 272 . . . . . 6 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → DECID (1st ‘(𝐽𝑚)) ∈ 𝑆)
15 eleq1 2178 . . . . . . . 8 (𝑛 = (2nd ‘(𝐽𝑚)) → (𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇 ↔ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
1615dcbid 806 . . . . . . 7 (𝑛 = (2nd ‘(𝐽𝑚)) → (DECID 𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇DECID (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
17 ctiunct.f . . . . . . . . . . 11 (𝜑𝐹:𝑆onto𝐴)
18 fof 5313 . . . . . . . . . . 11 (𝐹:𝑆onto𝐴𝐹:𝑆𝐴)
1917, 18syl 14 . . . . . . . . . 10 (𝜑𝐹:𝑆𝐴)
2019ad2antrr 477 . . . . . . . . 9 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → 𝐹:𝑆𝐴)
21 simpr 109 . . . . . . . . 9 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → (1st ‘(𝐽𝑚)) ∈ 𝑆)
2220, 21ffvelrnd 5522 . . . . . . . 8 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → (𝐹‘(1st ‘(𝐽𝑚))) ∈ 𝐴)
23 ctiunct.tdc . . . . . . . . . 10 ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)
2423ralrimiva 2480 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴𝑛 ∈ ω DECID 𝑛𝑇)
2524ad2antrr 477 . . . . . . . 8 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → ∀𝑥𝐴𝑛 ∈ ω DECID 𝑛𝑇)
26 nfcv 2256 . . . . . . . . . 10 𝑥ω
27 nfcsb1v 3003 . . . . . . . . . . . 12 𝑥(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇
2827nfcri 2250 . . . . . . . . . . 11 𝑥 𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇
2928nfdc 1620 . . . . . . . . . 10 𝑥DECID 𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇
3026, 29nfralya 2448 . . . . . . . . 9 𝑥𝑛 ∈ ω DECID 𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇
31 csbeq1a 2981 . . . . . . . . . . . 12 (𝑥 = (𝐹‘(1st ‘(𝐽𝑚))) → 𝑇 = (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)
3231eleq2d 2185 . . . . . . . . . . 11 (𝑥 = (𝐹‘(1st ‘(𝐽𝑚))) → (𝑛𝑇𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
3332dcbid 806 . . . . . . . . . 10 (𝑥 = (𝐹‘(1st ‘(𝐽𝑚))) → (DECID 𝑛𝑇DECID 𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
3433ralbidv 2412 . . . . . . . . 9 (𝑥 = (𝐹‘(1st ‘(𝐽𝑚))) → (∀𝑛 ∈ ω DECID 𝑛𝑇 ↔ ∀𝑛 ∈ ω DECID 𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
3530, 34rspc 2755 . . . . . . . 8 ((𝐹‘(1st ‘(𝐽𝑚))) ∈ 𝐴 → (∀𝑥𝐴𝑛 ∈ ω DECID 𝑛𝑇 → ∀𝑛 ∈ ω DECID 𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
3622, 25, 35sylc 62 . . . . . . 7 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → ∀𝑛 ∈ ω DECID 𝑛(𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)
3710adantr 272 . . . . . . . 8 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → (𝐽𝑚) ∈ (ω × ω))
38 xp2nd 6030 . . . . . . . 8 ((𝐽𝑚) ∈ (ω × ω) → (2nd ‘(𝐽𝑚)) ∈ ω)
3937, 38syl 14 . . . . . . 7 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → (2nd ‘(𝐽𝑚)) ∈ ω)
4016, 36, 39rspcdva 2766 . . . . . 6 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → DECID (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)
41 dcan 901 . . . . . 6 (DECID (1st ‘(𝐽𝑚)) ∈ 𝑆 → (DECID (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇DECID ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)))
4214, 40, 41sylc 62 . . . . 5 (((𝜑𝑚 ∈ ω) ∧ (1st ‘(𝐽𝑚)) ∈ 𝑆) → DECID ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
43 simpr 109 . . . . . . . 8 (((𝜑𝑚 ∈ ω) ∧ ¬ (1st ‘(𝐽𝑚)) ∈ 𝑆) → ¬ (1st ‘(𝐽𝑚)) ∈ 𝑆)
4443intnanrd 900 . . . . . . 7 (((𝜑𝑚 ∈ ω) ∧ ¬ (1st ‘(𝐽𝑚)) ∈ 𝑆) → ¬ ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
4544olcd 706 . . . . . 6 (((𝜑𝑚 ∈ ω) ∧ ¬ (1st ‘(𝐽𝑚)) ∈ 𝑆) → (((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇) ∨ ¬ ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)))
46 df-dc 803 . . . . . 6 (DECID ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇) ↔ (((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇) ∨ ¬ ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)))
4745, 46sylibr 133 . . . . 5 (((𝜑𝑚 ∈ ω) ∧ ¬ (1st ‘(𝐽𝑚)) ∈ 𝑆) → DECID ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
48 exmiddc 804 . . . . . 6 (DECID (1st ‘(𝐽𝑚)) ∈ 𝑆 → ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∨ ¬ (1st ‘(𝐽𝑚)) ∈ 𝑆))
4913, 48syl 14 . . . . 5 ((𝜑𝑚 ∈ ω) → ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∨ ¬ (1st ‘(𝐽𝑚)) ∈ 𝑆))
5042, 47, 49mpjaodan 770 . . . 4 ((𝜑𝑚 ∈ ω) → DECID ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
51 ibar 297 . . . . . . 7 (𝑚 ∈ ω → (((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇) ↔ (𝑚 ∈ ω ∧ ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))))
5251adantl 273 . . . . . 6 ((𝜑𝑚 ∈ ω) → (((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇) ↔ (𝑚 ∈ ω ∧ ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))))
53 2fveq3 5392 . . . . . . . . 9 (𝑧 = 𝑚 → (1st ‘(𝐽𝑧)) = (1st ‘(𝐽𝑚)))
5453eleq1d 2184 . . . . . . . 8 (𝑧 = 𝑚 → ((1st ‘(𝐽𝑧)) ∈ 𝑆 ↔ (1st ‘(𝐽𝑚)) ∈ 𝑆))
55 2fveq3 5392 . . . . . . . . 9 (𝑧 = 𝑚 → (2nd ‘(𝐽𝑧)) = (2nd ‘(𝐽𝑚)))
5653fveq2d 5391 . . . . . . . . . 10 (𝑧 = 𝑚 → (𝐹‘(1st ‘(𝐽𝑧))) = (𝐹‘(1st ‘(𝐽𝑚))))
5756csbeq1d 2979 . . . . . . . . 9 (𝑧 = 𝑚(𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇 = (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)
5855, 57eleq12d 2186 . . . . . . . 8 (𝑧 = 𝑚 → ((2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇 ↔ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇))
5954, 58anbi12d 462 . . . . . . 7 (𝑧 = 𝑚 → (((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇) ↔ ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)))
60 ctiunct.u . . . . . . 7 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}
6159, 60elrab2 2814 . . . . . 6 (𝑚𝑈 ↔ (𝑚 ∈ ω ∧ ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)))
6252, 61syl6rbbr 198 . . . . 5 ((𝜑𝑚 ∈ ω) → (𝑚𝑈 ↔ ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)))
6362dcbid 806 . . . 4 ((𝜑𝑚 ∈ ω) → (DECID 𝑚𝑈DECID ((1st ‘(𝐽𝑚)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑚)) ∈ (𝐹‘(1st ‘(𝐽𝑚))) / 𝑥𝑇)))
6450, 63mpbird 166 . . 3 ((𝜑𝑚 ∈ ω) → DECID 𝑚𝑈)
6564ralrimiva 2480 . 2 (𝜑 → ∀𝑚 ∈ ω DECID 𝑚𝑈)
66 eleq1 2178 . . . 4 (𝑚 = 𝑛 → (𝑚𝑈𝑛𝑈))
6766dcbid 806 . . 3 (𝑚 = 𝑛 → (DECID 𝑚𝑈DECID 𝑛𝑈))
6867cbvralv 2629 . 2 (∀𝑚 ∈ ω DECID 𝑚𝑈 ↔ ∀𝑛 ∈ ω DECID 𝑛𝑈)
6965, 68sylib 121 1 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑈)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  DECID wdc 802   = wceq 1314  wcel 1463  wral 2391  {crab 2395  csb 2973  wss 3039  ωcom 4472   × cxp 4505  wf 5087  ontowfo 5089  1-1-ontowf1o 5090  cfv 5091  1st c1st 6002  2nd c2nd 6003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005
This theorem is referenced by:  ctiunct  11848
  Copyright terms: Public domain W3C validator