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| Mirrors > Home > MPE Home > Th. List > oasuc | Structured version Visualization version GIF version | ||
| Description: Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. Definition 2.3 of [Schloeder] p. 4. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
| Ref | Expression |
|---|---|
| oasuc | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rdgsuc 8464 | . . 3 ⊢ (𝐵 ∈ On → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))) | |
| 2 | 1 | adantl 481 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))) |
| 3 | onsuc 7831 | . . 3 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On) | |
| 4 | oav 8549 | . . 3 ⊢ ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵)) | |
| 5 | 3, 4 | sylan2 593 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘suc 𝐵)) |
| 6 | ovex 7464 | . . . 4 ⊢ (𝐴 +o 𝐵) ∈ V | |
| 7 | suceq 6450 | . . . . 5 ⊢ (𝑥 = (𝐴 +o 𝐵) → suc 𝑥 = suc (𝐴 +o 𝐵)) | |
| 8 | eqid 2737 | . . . . 5 ⊢ (𝑥 ∈ V ↦ suc 𝑥) = (𝑥 ∈ V ↦ suc 𝑥) | |
| 9 | 6 | sucex 7826 | . . . . 5 ⊢ suc (𝐴 +o 𝐵) ∈ V |
| 10 | 7, 8, 9 | fvmpt 7016 | . . . 4 ⊢ ((𝐴 +o 𝐵) ∈ V → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵)) |
| 11 | 6, 10 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +o 𝐵)) = suc (𝐴 +o 𝐵) |
| 12 | oav 8549 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵)) | |
| 13 | 12 | fveq2d 6910 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ V ↦ suc 𝑥)‘(𝐴 +o 𝐵)) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))) |
| 14 | 11, 13 | eqtr3id 2791 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 +o 𝐵) = ((𝑥 ∈ V ↦ suc 𝑥)‘(rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))) |
| 15 | 2, 5, 14 | 3eqtr4d 2787 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 Oncon0 6384 suc csuc 6386 ‘cfv 6561 (class class class)co 7431 reccrdg 8449 +o coa 8503 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-oadd 8510 |
| This theorem is referenced by: oacl 8573 oa0r 8576 oaordi 8584 oawordri 8588 oawordeulem 8592 oalimcl 8598 oaass 8599 oarec 8600 odi 8617 oeoalem 8634 oa0suclim 43288 dflim5 43342 naddgeoa 43407 naddonnn 43408 naddwordnexlem4 43414 |
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