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Mirrors > Home > MPE Home > Th. List > oav | Structured version Visualization version GIF version |
Description: Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oav | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgeq2 8396 | . . 3 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦) = rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)) | |
2 | 1 | fveq1d 6881 | . 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧)) |
3 | fveq2 6879 | . 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)) | |
4 | df-oadd 8454 | . 2 ⊢ +o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧)) | |
5 | fvex 6892 | . 2 ⊢ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵) ∈ V | |
6 | 2, 3, 4, 5 | ovmpo 7552 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ↦ cmpt 5225 Oncon0 6354 suc csuc 6356 ‘cfv 6533 (class class class)co 7394 reccrdg 8393 +o coa 8447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-oadd 8454 |
This theorem is referenced by: oa0 8500 oasuc 8508 onasuc 8512 oalim 8516 |
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