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Mirrors > Home > MPE Home > Th. List > oalim | Structured version Visualization version GIF version |
Description: Ordinal addition with a limit ordinal. Definition 8.1 of [TakeutiZaring] p. 56. Definition 2.3 of [Schloeder] p. 4. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oalim | ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limelon 6450 | . . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On) | |
2 | simpr 484 | . . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → Lim 𝐵) | |
3 | 1, 2 | jca 511 | . 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵)) |
4 | rdglim2a 8472 | . . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)) | |
5 | 4 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)) |
6 | oav 8548 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵)) | |
7 | onelon 6411 | . . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On) | |
8 | oav 8548 | . . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)) | |
9 | 7, 8 | sylan2 593 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ 𝐵)) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)) |
10 | 9 | anassrs 467 | . . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥 ∈ 𝐵) → (𝐴 +o 𝑥) = (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)) |
11 | 10 | iuneq2dv 5021 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥)) |
12 | 6, 11 | eqeq12d 2751 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥) ↔ (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))) |
13 | 12 | adantrr 717 | . . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → ((𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥) ↔ (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec((𝑦 ∈ V ↦ suc 𝑦), 𝐴)‘𝑥))) |
14 | 5, 13 | mpbird 257 | . 2 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥)) |
15 | 3, 14 | sylan2 593 | 1 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 +o 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ ciun 4996 ↦ cmpt 5231 Oncon0 6386 Lim wlim 6387 suc csuc 6388 ‘cfv 6563 (class class class)co 7431 reccrdg 8448 +o coa 8502 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-oadd 8509 |
This theorem is referenced by: oacl 8572 oa0r 8575 oaordi 8583 oawordri 8587 oawordeulem 8591 oalimcl 8597 oaass 8598 oarec 8599 odi 8616 oeoalem 8633 oaabslem 8684 oaabs2 8686 oa0suclim 43265 naddgeoa 43384 naddwordnexlem4 43391 |
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