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Mirrors > Home > MPE Home > Th. List > om0 | Structured version Visualization version GIF version |
Description: Ordinal multiplication with zero. Definition 8.15(a) of [TakeutiZaring] p. 62. See om0x 8433 for a way to remove the antecedent 𝐴 ∈ On. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
om0 | ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6368 | . . 3 ⊢ ∅ ∈ On | |
2 | omv 8426 | . . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅)) | |
3 | 1, 2 | mpan2 690 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅)) |
4 | 0ex 5263 | . . 3 ⊢ ∅ ∈ V | |
5 | 4 | rdg0 8335 | . 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅) = ∅ |
6 | 3, 5 | eqtrdi 2794 | 1 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4281 ↦ cmpt 5187 Oncon0 6314 ‘cfv 6492 (class class class)co 7350 reccrdg 8323 +o coa 8377 ·o comu 8378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-omul 8385 |
This theorem is referenced by: om0x 8433 oesuclem 8439 omcl 8450 om0r 8453 om1 8457 om1r 8458 omwordri 8487 om00 8490 odi 8494 omass 8495 omeulem1 8497 oen0 8501 oeoa 8512 oeoelem 8513 oeeui 8517 nnm0 8520 nnm0r 8525 nneob 8570 cantnfle 9541 cantnfp1 9551 fin1a2lem6 10275 dflim5 41457 omcl3g 41461 |
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