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Mirrors > Home > MPE Home > Th. List > omv | Structured version Visualization version GIF version |
Description: Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
omv | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 +𝑜 𝑦) = (𝑥 +𝑜 𝐴)) | |
2 | 1 | mpteq2dv 4673 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))) |
3 | rdgeq1 7394 | . . . 4 ⊢ ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅) = rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅) = rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)) |
5 | 4 | fveq1d 6105 | . 2 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝑧)) |
6 | fveq2 6103 | . 2 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵)) | |
7 | df-omul 7452 | . 2 ⊢ ·𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅)‘𝑧)) | |
8 | fvex 6113 | . 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ V | |
9 | 5, 6, 7, 8 | ovmpt2 6694 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ↦ cmpt 4643 Oncon0 5640 ‘cfv 5804 (class class class)co 6549 reccrdg 7392 +𝑜 coa 7444 ·𝑜 comu 7445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-omul 7452 |
This theorem is referenced by: om0 7484 omsuc 7493 onmsuc 7496 omlim 7500 |
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