MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnfcom2 Structured version   Visualization version   GIF version

Theorem cnfcom2 8543
Description: Any nonzero ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
Hypotheses
Ref Expression
cnfcom.s 𝑆 = dom (ω CNF 𝐴)
cnfcom.a (𝜑𝐴 ∈ On)
cnfcom.b (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))
cnfcom.f 𝐹 = ((ω CNF 𝐴)‘𝐵)
cnfcom.g 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom.h 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
cnfcom.t 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom.m 𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
cnfcom.k 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
cnfcom.w 𝑊 = (𝐺 dom 𝐺)
cnfcom2.1 (𝜑 → ∅ ∈ 𝐵)
Assertion
Ref Expression
cnfcom2 (𝜑 → (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
Distinct variable groups:   𝑥,𝑘,𝑧,𝐴   𝑥,𝑀   𝑓,𝑘,𝑥,𝑧,𝐹   𝑧,𝑇   𝑥,𝑊   𝑓,𝐺,𝑘,𝑥,𝑧   𝑓,𝐻,𝑥   𝑆,𝑘,𝑧   𝜑,𝑘,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓)   𝐵(𝑥,𝑧,𝑓,𝑘)   𝑆(𝑥,𝑓)   𝑇(𝑥,𝑓,𝑘)   𝐻(𝑧,𝑘)   𝐾(𝑥,𝑧,𝑓,𝑘)   𝑀(𝑧,𝑓,𝑘)   𝑊(𝑧,𝑓,𝑘)

Proof of Theorem cnfcom2
StepHypRef Expression
1 cnfcom.s . . . . 5 𝑆 = dom (ω CNF 𝐴)
2 cnfcom.a . . . . 5 (𝜑𝐴 ∈ On)
3 cnfcom.b . . . . 5 (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))
4 cnfcom.f . . . . 5 𝐹 = ((ω CNF 𝐴)‘𝐵)
5 cnfcom.g . . . . 5 𝐺 = OrdIso( E , (𝐹 supp ∅))
6 cnfcom.h . . . . 5 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
7 cnfcom.t . . . . 5 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
8 cnfcom.m . . . . 5 𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
9 cnfcom.k . . . . 5 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
10 ovex 6632 . . . . . . . . . 10 (𝐹 supp ∅) ∈ V
115oion 8385 . . . . . . . . . 10 ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
1210, 11ax-mp 5 . . . . . . . . 9 dom 𝐺 ∈ On
1312elexi 3199 . . . . . . . 8 dom 𝐺 ∈ V
1413uniex 6906 . . . . . . 7 dom 𝐺 ∈ V
1514sucid 5763 . . . . . 6 dom 𝐺 ∈ suc dom 𝐺
16 cnfcom.w . . . . . . 7 𝑊 = (𝐺 dom 𝐺)
17 cnfcom2.1 . . . . . . 7 (𝜑 → ∅ ∈ 𝐵)
181, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17cnfcom2lem 8542 . . . . . 6 (𝜑 → dom 𝐺 = suc dom 𝐺)
1915, 18syl5eleqr 2705 . . . . 5 (𝜑 dom 𝐺 ∈ dom 𝐺)
201, 2, 3, 4, 5, 6, 7, 8, 9, 19cnfcom 8541 . . . 4 (𝜑 → (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺))))
2116oveq2i 6615 . . . . . 6 (ω ↑𝑜 𝑊) = (ω ↑𝑜 (𝐺 dom 𝐺))
2216fveq2i 6151 . . . . . 6 (𝐹𝑊) = (𝐹‘(𝐺 dom 𝐺))
2321, 22oveq12i 6616 . . . . 5 ((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) = ((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺)))
24 f1oeq3 6086 . . . . 5 (((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) = ((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺))) → ((𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺)))))
2523, 24ax-mp 5 . . . 4 ((𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺))))
2620, 25sylibr 224 . . 3 (𝜑 → (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
2718fveq2d 6152 . . . 4 (𝜑 → (𝑇‘dom 𝐺) = (𝑇‘suc dom 𝐺))
28 f1oeq1 6084 . . . 4 ((𝑇‘dom 𝐺) = (𝑇‘suc dom 𝐺) → ((𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))))
2927, 28syl 17 . . 3 (𝜑 → ((𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))))
3026, 29mpbird 247 . 2 (𝜑 → (𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
314fveq2i 6151 . . . . 5 ((ω CNF 𝐴)‘𝐹) = ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵))
32 omelon 8487 . . . . . . 7 ω ∈ On
3332a1i 11 . . . . . 6 (𝜑 → ω ∈ On)
341, 33, 2cantnff1o 8537 . . . . . . . . 9 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
35 f1ocnv 6106 . . . . . . . . 9 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
36 f1of 6094 . . . . . . . . 9 ((ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
3734, 35, 363syl 18 . . . . . . . 8 (𝜑(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
3837, 3ffvelrnd 6316 . . . . . . 7 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
394, 38syl5eqel 2702 . . . . . 6 (𝜑𝐹𝑆)
408oveq1i 6614 . . . . . . . . . 10 (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)
4140a1i 11 . . . . . . . . 9 ((𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
4241mpt2eq3ia 6673 . . . . . . . 8 (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
43 eqid 2621 . . . . . . . 8 ∅ = ∅
44 seqomeq12 7494 . . . . . . . 8 (((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)) ∧ ∅ = ∅) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅))
4542, 43, 44mp2an 707 . . . . . . 7 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
466, 45eqtri 2643 . . . . . 6 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
471, 33, 2, 5, 39, 46cantnfval 8509 . . . . 5 (𝜑 → ((ω CNF 𝐴)‘𝐹) = (𝐻‘dom 𝐺))
4831, 47syl5reqr 2670 . . . 4 (𝜑 → (𝐻‘dom 𝐺) = ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)))
4918fveq2d 6152 . . . 4 (𝜑 → (𝐻‘dom 𝐺) = (𝐻‘suc dom 𝐺))
50 f1ocnvfv2 6487 . . . . 5 (((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) ∧ 𝐵 ∈ (ω ↑𝑜 𝐴)) → ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)) = 𝐵)
5134, 3, 50syl2anc 692 . . . 4 (𝜑 → ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)) = 𝐵)
5248, 49, 513eqtr3d 2663 . . 3 (𝜑 → (𝐻‘suc dom 𝐺) = 𝐵)
53 f1oeq2 6085 . . 3 ((𝐻‘suc dom 𝐺) = 𝐵 → ((𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))))
5452, 53syl 17 . 2 (𝜑 → ((𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))))
5530, 54mpbid 222 1 (𝜑 → (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cun 3553  c0 3891   cuni 4402  cmpt 4673   E cep 4983  ccnv 5073  dom cdm 5074  Oncon0 5682  suc csuc 5684  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  cmpt2 6606  ωcom 7012   supp csupp 7240  seq𝜔cseqom 7487   +𝑜 coa 7502   ·𝑜 comu 7503  𝑜 coe 7504  OrdIsocoi 8358   CNF ccnf 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-cnf 8503
This theorem is referenced by:  cnfcom3  8545
  Copyright terms: Public domain W3C validator