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Theorem cnfcom2 8767
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 6827 . . . . . . . . . 10 (𝐹 supp ∅) ∈ V
115oion 8601 . . . . . . . . . 10 ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
1210, 11ax-mp 5 . . . . . . . . 9 dom 𝐺 ∈ On
1312elexi 3365 . . . . . . . 8 dom 𝐺 ∈ V
1413uniex 7104 . . . . . . 7 dom 𝐺 ∈ V
1514sucid 5946 . . . . . 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 8766 . . . . . 6 (𝜑 → dom 𝐺 = suc dom 𝐺)
1915, 18syl5eleqr 2857 . . . . 5 (𝜑 dom 𝐺 ∈ dom 𝐺)
201, 2, 3, 4, 5, 6, 7, 8, 9, 19cnfcom 8765 . . . 4 (𝜑 → (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺))))
2116oveq2i 6807 . . . . . 6 (ω ↑𝑜 𝑊) = (ω ↑𝑜 (𝐺 dom 𝐺))
2216fveq2i 6336 . . . . . 6 (𝐹𝑊) = (𝐹‘(𝐺 dom 𝐺))
2321, 22oveq12i 6808 . . . . 5 ((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) = ((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺)))
24 f1oeq3 6271 . . . . 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 6337 . . . 4 (𝜑 → (𝑇‘dom 𝐺) = (𝑇‘suc dom 𝐺))
28 f1oeq1 6269 . . . 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 6336 . . . . 5 ((ω CNF 𝐴)‘𝐹) = ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵))
32 omelon 8711 . . . . . . 7 ω ∈ On
3332a1i 11 . . . . . 6 (𝜑 → ω ∈ On)
341, 33, 2cantnff1o 8761 . . . . . . . . 9 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
35 f1ocnv 6291 . . . . . . . . 9 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
36 f1of 6279 . . . . . . . . 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 6505 . . . . . . 7 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
394, 38syl5eqel 2854 . . . . . 6 (𝜑𝐹𝑆)
408oveq1i 6806 . . . . . . . . . 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 6871 . . . . . . . 8 (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
43 eqid 2771 . . . . . . . 8 ∅ = ∅
44 seqomeq12 7706 . . . . . . . 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 672 . . . . . . 7 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
466, 45eqtri 2793 . . . . . 6 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
471, 33, 2, 5, 39, 46cantnfval 8733 . . . . 5 (𝜑 → ((ω CNF 𝐴)‘𝐹) = (𝐻‘dom 𝐺))
4831, 47syl5reqr 2820 . . . 4 (𝜑 → (𝐻‘dom 𝐺) = ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)))
4918fveq2d 6337 . . . 4 (𝜑 → (𝐻‘dom 𝐺) = (𝐻‘suc dom 𝐺))
50 f1ocnvfv2 6679 . . . . 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 573 . . . 4 (𝜑 → ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)) = 𝐵)
5248, 49, 513eqtr3d 2813 . . 3 (𝜑 → (𝐻‘suc dom 𝐺) = 𝐵)
53 f1oeq2 6270 . . 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 382   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  c0 4063   cuni 4575  cmpt 4864   E cep 5162  ccnv 5249  dom cdm 5250  Oncon0 5865  suc csuc 5867  wf 6026  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6796  cmpt2 6798  ωcom 7216   supp csupp 7450  seq𝜔cseqom 7699   +𝑜 coa 7714   ·𝑜 comu 7715  𝑜 coe 7716  OrdIsocoi 8574   CNF ccnf 8726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-inf2 8706
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-supp 7451  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-seqom 7700  df-1o 7717  df-2o 7718  df-oadd 7721  df-omul 7722  df-oexp 7723  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fsupp 8436  df-oi 8575  df-cnf 8727
This theorem is referenced by:  cnfcom3  8769
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