MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnflem4 Structured version   Visualization version   GIF version

Theorem cantnflem4 8886
Description: Lemma for cantnf 8887. Complete the induction step of cantnflem3 8885. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
cantnf.c (𝜑𝐶 ∈ (𝐴o 𝐵))
cantnf.s (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))
cantnf.e (𝜑 → ∅ ∈ 𝐶)
cantnf.x 𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴o 𝑐)}
cantnf.p 𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
cantnf.y 𝑌 = (1st𝑃)
cantnf.z 𝑍 = (2nd𝑃)
Assertion
Ref Expression
cantnflem4 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Distinct variable groups:   𝑤,𝑐,𝑥,𝑦,𝑧,𝐵   𝑎,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧,𝐶   𝐴,𝑎,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐   𝑆,𝑐,𝑥,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑤,𝑌,𝑥,𝑦,𝑧   𝑋,𝑎,𝑏,𝑑,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑤,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑎,𝑏,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏,𝑐,𝑑)   𝑆(𝑤,𝑎,𝑏,𝑑)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏,𝑑)   𝑋(𝑐)   𝑌(𝑎,𝑏,𝑐,𝑑)   𝑍(𝑤,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem cantnflem4
Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnf.s . . . 4 (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))
2 cantnfs.a . . . . . . . . 9 (𝜑𝐴 ∈ On)
3 cantnfs.s . . . . . . . . . . . . 13 𝑆 = dom (𝐴 CNF 𝐵)
4 cantnfs.b . . . . . . . . . . . . 13 (𝜑𝐵 ∈ On)
5 oemapval.t . . . . . . . . . . . . 13 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
6 cantnf.c . . . . . . . . . . . . 13 (𝜑𝐶 ∈ (𝐴o 𝐵))
7 cantnf.e . . . . . . . . . . . . 13 (𝜑 → ∅ ∈ 𝐶)
83, 2, 4, 5, 6, 1, 7cantnflem2 8884 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
9 eqid 2778 . . . . . . . . . . . . . 14 𝑋 = 𝑋
10 eqid 2778 . . . . . . . . . . . . . 14 𝑌 = 𝑌
11 eqid 2778 . . . . . . . . . . . . . 14 𝑍 = 𝑍
129, 10, 113pm3.2i 1395 . . . . . . . . . . . . 13 (𝑋 = 𝑋𝑌 = 𝑌𝑍 = 𝑍)
13 cantnf.x . . . . . . . . . . . . . 14 𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴o 𝑐)}
14 cantnf.p . . . . . . . . . . . . . 14 𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
15 cantnf.y . . . . . . . . . . . . . 14 𝑌 = (1st𝑃)
16 cantnf.z . . . . . . . . . . . . . 14 𝑍 = (2nd𝑃)
1713, 14, 15, 16oeeui 7966 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → (((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)) ∧ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶) ↔ (𝑋 = 𝑋𝑌 = 𝑌𝑍 = 𝑍)))
1812, 17mpbiri 250 . . . . . . . . . . . 12 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)) ∧ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
198, 18syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)) ∧ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
2019simpld 490 . . . . . . . . . 10 (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)))
2120simp1d 1133 . . . . . . . . 9 (𝜑𝑋 ∈ On)
22 oecl 7901 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
232, 21, 22syl2anc 579 . . . . . . . 8 (𝜑 → (𝐴o 𝑋) ∈ On)
2420simp2d 1134 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝐴 ∖ 1o))
2524eldifad 3804 . . . . . . . . 9 (𝜑𝑌𝐴)
26 onelon 6001 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
272, 25, 26syl2anc 579 . . . . . . . 8 (𝜑𝑌 ∈ On)
28 omcl 7900 . . . . . . . 8 (((𝐴o 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
2923, 27, 28syl2anc 579 . . . . . . 7 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
3020simp3d 1135 . . . . . . . 8 (𝜑𝑍 ∈ (𝐴o 𝑋))
31 onelon 6001 . . . . . . . 8 (((𝐴o 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴o 𝑋)) → 𝑍 ∈ On)
3223, 30, 31syl2anc 579 . . . . . . 7 (𝜑𝑍 ∈ On)
33 oaword1 7916 . . . . . . 7 ((((𝐴o 𝑋) ·o 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴o 𝑋) ·o 𝑌) ⊆ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
3429, 32, 33syl2anc 579 . . . . . 6 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ⊆ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
35 dif1o 7864 . . . . . . . . . . 11 (𝑌 ∈ (𝐴 ∖ 1o) ↔ (𝑌𝐴𝑌 ≠ ∅))
3635simprbi 492 . . . . . . . . . 10 (𝑌 ∈ (𝐴 ∖ 1o) → 𝑌 ≠ ∅)
3724, 36syl 17 . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
38 on0eln0 6031 . . . . . . . . . 10 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
3927, 38syl 17 . . . . . . . . 9 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
4037, 39mpbird 249 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑌)
41 omword1 7937 . . . . . . . 8 ((((𝐴o 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴o 𝑋) ⊆ ((𝐴o 𝑋) ·o 𝑌))
4223, 27, 40, 41syl21anc 828 . . . . . . 7 (𝜑 → (𝐴o 𝑋) ⊆ ((𝐴o 𝑋) ·o 𝑌))
4342, 30sseldd 3822 . . . . . 6 (𝜑𝑍 ∈ ((𝐴o 𝑋) ·o 𝑌))
4434, 43sseldd 3822 . . . . 5 (𝜑𝑍 ∈ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
4519simprd 491 . . . . 5 (𝜑 → (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶)
4644, 45eleqtrd 2861 . . . 4 (𝜑𝑍𝐶)
471, 46sseldd 3822 . . 3 (𝜑𝑍 ∈ ran (𝐴 CNF 𝐵))
483, 2, 4cantnff 8868 . . . 4 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
49 ffn 6291 . . . 4 ((𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
50 fvelrnb 6503 . . . 4 ((𝐴 CNF 𝐵) Fn 𝑆 → (𝑍 ∈ ran (𝐴 CNF 𝐵) ↔ ∃𝑔𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍))
5148, 49, 503syl 18 . . 3 (𝜑 → (𝑍 ∈ ran (𝐴 CNF 𝐵) ↔ ∃𝑔𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍))
5247, 51mpbid 224 . 2 (𝜑 → ∃𝑔𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
532adantr 474 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐴 ∈ On)
544adantr 474 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐵 ∈ On)
556adantr 474 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ (𝐴o 𝐵))
561adantr 474 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
577adantr 474 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ∅ ∈ 𝐶)
58 simprl 761 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝑔𝑆)
59 simprr 763 . . 3 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
60 eqid 2778 . . 3 (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝑔𝑡))) = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝑔𝑡)))
613, 53, 54, 5, 55, 56, 57, 13, 14, 15, 16, 58, 59, 60cantnflem3 8885 . 2 ((𝜑 ∧ (𝑔𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ ran (𝐴 CNF 𝐵))
6252, 61rexlimddv 3218 1 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969  wral 3090  wrex 3091  {crab 3094  cdif 3789  wss 3792  c0 4141  ifcif 4307  cop 4404   cuni 4671   cint 4710  {copab 4948  cmpt 4965  dom cdm 5355  ran crn 5356  Oncon0 5976  cio 6097   Fn wfn 6130  wf 6131  cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  1oc1o 7836  2oc2o 7837   +o coa 7840   ·o comu 7841  o coe 7842   CNF ccnf 8855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-seqom 7826  df-1o 7843  df-2o 7844  df-oadd 7847  df-omul 7848  df-oexp 7849  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-oi 8704  df-cnf 8856
This theorem is referenced by:  cantnf  8887
  Copyright terms: Public domain W3C validator