Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > oa0 | Structured version Visualization version GIF version |
Description: Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oa0 | ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6247 | . . 3 ⊢ ∅ ∈ On | |
2 | oav 8139 | . . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 +o ∅) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅)) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅)) |
4 | rdg0g 8066 | . 2 ⊢ (𝐴 ∈ On → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘∅) = 𝐴) | |
5 | 3, 4 | eqtrd 2859 | 1 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 ↦ cmpt 5149 Oncon0 6194 suc csuc 6196 ‘cfv 6358 (class class class)co 7159 reccrdg 8048 +o coa 8102 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-oadd 8109 |
This theorem is referenced by: oa1suc 8159 oacl 8163 oa0r 8166 om0r 8167 oawordri 8179 oaord1 8180 oaword1 8181 oawordeulem 8183 oa00 8188 oaass 8190 oarec 8191 odi 8208 oeoalem 8225 nna0 8233 nna0r 8238 nnm0r 8239 nnawordi 8250 cantnflt 9138 rdgeqoa 34655 |
Copyright terms: Public domain | W3C validator |