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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. Definition 2.5 of [Schloeder] p. 4. See om0x 8556 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 6440 | . . 3 ⊢ ∅ ∈ On | |
2 | omv 8549 | . . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅)) | |
3 | 1, 2 | mpan2 691 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅)) |
4 | 0ex 5313 | . . 3 ⊢ ∅ ∈ V | |
5 | 4 | rdg0 8460 | . 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘∅) = ∅ |
6 | 3, 5 | eqtrdi 2791 | 1 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ↦ cmpt 5231 Oncon0 6386 ‘cfv 6563 (class class class)co 7431 reccrdg 8448 +o coa 8502 ·o comu 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-omul 8510 |
This theorem is referenced by: om0x 8556 oesuclem 8562 omcl 8573 om0r 8576 om1 8579 om1r 8580 omwordri 8609 om00 8612 odi 8616 omass 8617 omeulem1 8619 oen0 8623 oeoa 8634 oeoelem 8635 oeeui 8639 nnm0 8642 nnm0r 8647 nneob 8693 cantnfle 9709 cantnfp1 9719 fin1a2lem6 10443 onexlimgt 43232 om0suclim 43266 oaabsb 43284 dflim5 43319 onmcl 43321 omcl3g 43324 |
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