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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 8533 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
4 | 1, 2, 3 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
5 | eloni 6365 | . . . 4 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵)) | |
6 | ordwe 6368 | . . . 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 9685 | . . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵))) |
11 | isowe 7339 | . . . 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 257 | . 2 ⊢ (𝜑 → 𝑇 We 𝑆) |
14 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑o 𝐵)) |
15 | isocnv 7320 | . . . . . 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 7435 | . . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V | |
18 | 17 | dmex 7896 | . . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V |
19 | 8, 18 | eqeltri 2821 | . . . . . . 7 ⊢ 𝑆 ∈ V |
20 | exse 5630 | . . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆) | |
21 | 19, 20 | ax-mp 5 | . . . . . 6 ⊢ 𝑇 Se 𝑆 |
22 | eqid 2724 | . . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆) | |
23 | 22 | oieu 9531 | . . . . . 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 585 | . . . . 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 709 | . . . 4 ⊢ (𝜑 → ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))) |
26 | 25 | simpld 494 | . . 3 ⊢ (𝜑 → (𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆)) |
27 | 26 | eqcomd 2730 | . 2 ⊢ (𝜑 → dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵)) |
28 | 13, 27 | jca 511 | 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 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 Vcvv 3466 {copab 5201 E cep 5570 Se wse 5620 We wwe 5621 ◡ccnv 5666 dom cdm 5667 Ord word 6354 Oncon0 6355 ‘cfv 6534 Isom wiso 6535 (class class class)co 7402 ↑o coe 8461 OrdIsocoi 9501 CNF ccnf 9653 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-seqom 8444 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-oexp 8468 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-oi 9502 df-cnf 9654 |
This theorem is referenced by: cantnffval2 9687 wemapwe 9689 |
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