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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 8156 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
4 | 1, 2, 3 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
5 | eloni 6196 | . . . 4 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵)) | |
6 | ordwe 6199 | . . . 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 9150 | . . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵))) |
11 | isowe 7096 | . . . 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 259 | . 2 ⊢ (𝜑 → 𝑇 We 𝑆) |
14 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑o 𝐵)) |
15 | isocnv 7077 | . . . . . 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 7183 | . . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V | |
18 | 17 | dmex 7610 | . . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V |
19 | 8, 18 | eqeltri 2909 | . . . . . . 7 ⊢ 𝑆 ∈ V |
20 | exse 5514 | . . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆) | |
21 | 19, 20 | ax-mp 5 | . . . . . 6 ⊢ 𝑇 Se 𝑆 |
22 | eqid 2821 | . . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆) | |
23 | 22 | oieu 8997 | . . . . . 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 588 | . . . . 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 497 | . . 3 ⊢ (𝜑 → (𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆)) |
27 | 26 | eqcomd 2827 | . 2 ⊢ (𝜑 → dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵)) |
28 | 13, 27 | jca 514 | 1 ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3495 {copab 5121 E cep 5459 Se wse 5507 We wwe 5508 ◡ccnv 5549 dom cdm 5550 Ord word 6185 Oncon0 6186 ‘cfv 6350 Isom wiso 6351 (class class class)co 7150 ↑o coe 8095 OrdIsocoi 8967 CNF ccnf 9118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-seqom 8078 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-oexp 8102 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-cnf 9119 |
This theorem is referenced by: cantnffval2 9152 wemapwe 9154 |
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