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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 8416 | . . 3 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦) = rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)) | |
2 | 1 | fveq1d 6893 | . 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧)) |
3 | fveq2 6891 | . 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝑧) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)) | |
4 | df-oadd 8474 | . 2 ⊢ +o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝑦)‘𝑧)) | |
5 | fvex 6904 | . 2 ⊢ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵) ∈ V | |
6 | 2, 3, 4, 5 | ovmpo 7571 | 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 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ↦ cmpt 5231 Oncon0 6364 suc csuc 6366 ‘cfv 6543 (class class class)co 7412 reccrdg 8413 +o coa 8467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-oadd 8474 |
This theorem is referenced by: oa0 8520 oasuc 8528 onasuc 8532 oalim 8536 |
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