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Mirrors > Home > MPE Home > Th. List > oemapwe | Structured version Visualization version GIF version |
Description: The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternate definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.) |
Ref | Expression |
---|---|
cantnfs.s | ⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
cantnfs.a | ⊢ (𝜑 → 𝐴 ∈ On) |
cantnfs.b | ⊢ (𝜑 → 𝐵 ∈ On) |
oemapval.t | ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} |
Ref | Expression |
---|---|
oemapwe | ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cantnfs.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ On) | |
2 | cantnfs.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ On) | |
3 | oecl 8558 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
4 | 1, 2, 3 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
5 | eloni 6381 | . . . 4 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵)) | |
6 | ordwe 6384 | . . . 4 ⊢ (Ord (𝐴 ↑o 𝐵) → E We (𝐴 ↑o 𝐵)) | |
7 | 4, 5, 6 | 3syl 18 | . . 3 ⊢ (𝜑 → E We (𝐴 ↑o 𝐵)) |
8 | cantnfs.s | . . . . 5 ⊢ 𝑆 = dom (𝐴 CNF 𝐵) | |
9 | oemapval.t | . . . . 5 ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} | |
10 | 8, 1, 2, 9 | cantnf 9718 | . . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵))) |
11 | isowe 7356 | . . . 4 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → (𝑇 We 𝑆 ↔ E We (𝐴 ↑o 𝐵))) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → (𝑇 We 𝑆 ↔ E We (𝐴 ↑o 𝐵))) |
13 | 7, 12 | mpbird 256 | . 2 ⊢ (𝜑 → 𝑇 We 𝑆) |
14 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑o 𝐵)) |
15 | isocnv 7337 | . . . . . 6 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) | |
16 | 10, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) |
17 | ovex 7452 | . . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V | |
18 | 17 | dmex 7917 | . . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V |
19 | 8, 18 | eqeltri 2821 | . . . . . . 7 ⊢ 𝑆 ∈ V |
20 | exse 5641 | . . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆) | |
21 | 19, 20 | ax-mp 5 | . . . . . 6 ⊢ 𝑇 Se 𝑆 |
22 | eqid 2725 | . . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆) | |
23 | 22 | oieu 9564 | . . . . . 6 ⊢ ((𝑇 We 𝑆 ∧ 𝑇 Se 𝑆) → ((Ord (𝐴 ↑o 𝐵) ∧ ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) ↔ ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))) |
24 | 13, 21, 23 | sylancl 584 | . . . . 5 ⊢ (𝜑 → ((Ord (𝐴 ↑o 𝐵) ∧ ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) ↔ ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))) |
25 | 14, 16, 24 | mpbi2and 710 | . . . 4 ⊢ (𝜑 → ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))) |
26 | 25 | simpld 493 | . . 3 ⊢ (𝜑 → (𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆)) |
27 | 26 | eqcomd 2731 | . 2 ⊢ (𝜑 → dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵)) |
28 | 13, 27 | jca 510 | 1 ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 Vcvv 3461 {copab 5211 E cep 5581 Se wse 5631 We wwe 5632 ◡ccnv 5677 dom cdm 5678 Ord word 6370 Oncon0 6371 ‘cfv 6549 Isom wiso 6550 (class class class)co 7419 ↑o coe 8486 OrdIsocoi 9534 CNF ccnf 9686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-seqom 8469 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-oexp 8493 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-oi 9535 df-cnf 9687 |
This theorem is referenced by: cantnffval2 9720 wemapwe 9722 |
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