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Mirrors > Home > MPE Home > Th. List > o2p2e4 | Structured version Visualization version GIF version |
Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6190. For the usual proof using complex numbers, see 2p2e4 11760. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5181, from a comment by Sophie. (Revised by SN, 23-Mar-2024.) |
Ref | Expression |
---|---|
o2p2e4 | ⊢ (2o +o 2o) = 4o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on 8100 | . . . 4 ⊢ 2o ∈ On | |
2 | df-1o 8091 | . . . . 5 ⊢ 1o = suc ∅ | |
3 | peano1 7590 | . . . . . 6 ⊢ ∅ ∈ ω | |
4 | peano2 7591 | . . . . . 6 ⊢ (∅ ∈ ω → suc ∅ ∈ ω) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ suc ∅ ∈ ω |
6 | 2, 5 | eqeltri 2906 | . . . 4 ⊢ 1o ∈ ω |
7 | onasuc 8142 | . . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ ω) → (2o +o suc 1o) = suc (2o +o 1o)) | |
8 | 1, 6, 7 | mp2an 688 | . . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o) |
9 | df-2o 8092 | . . . 4 ⊢ 2o = suc 1o | |
10 | 9 | oveq2i 7156 | . . 3 ⊢ (2o +o 2o) = (2o +o suc 1o) |
11 | df-3o 8093 | . . . . 5 ⊢ 3o = suc 2o | |
12 | oa1suc 8145 | . . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o) | |
13 | 1, 12 | ax-mp 5 | . . . . 5 ⊢ (2o +o 1o) = suc 2o |
14 | 11, 13 | eqtr4i 2844 | . . . 4 ⊢ 3o = (2o +o 1o) |
15 | suceq 6249 | . . . 4 ⊢ (3o = (2o +o 1o) → suc 3o = suc (2o +o 1o)) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ suc 3o = suc (2o +o 1o) |
17 | 8, 10, 16 | 3eqtr4i 2851 | . 2 ⊢ (2o +o 2o) = suc 3o |
18 | df-4o 8094 | . 2 ⊢ 4o = suc 3o | |
19 | 17, 18 | eqtr4i 2844 | 1 ⊢ (2o +o 2o) = 4o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ∅c0 4288 Oncon0 6184 suc csuc 6186 (class class class)co 7145 ωcom 7569 1oc1o 8084 2oc2o 8085 3oc3o 8086 4oc4o 8087 +o coa 8088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-3o 8093 df-4o 8094 df-oadd 8095 |
This theorem is referenced by: (None) |
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