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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 5970. For the usual proof using complex numbers, see 2p2e4 11494. (Contributed by NM, 18-Aug-2021.) |
Ref | Expression |
---|---|
o2p2e4 | ⊢ (2o +o 2o) = 4o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on 7836 | . . . 4 ⊢ 2o ∈ On | |
2 | 1on 7834 | . . . 4 ⊢ 1o ∈ On | |
3 | oasuc 7872 | . . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ On) → (2o +o suc 1o) = suc (2o +o 1o)) | |
4 | 1, 2, 3 | mp2an 685 | . . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o) |
5 | df-2o 7828 | . . . 4 ⊢ 2o = suc 1o | |
6 | 5 | oveq2i 6917 | . . 3 ⊢ (2o +o 2o) = (2o +o suc 1o) |
7 | df-3o 7829 | . . . . 5 ⊢ 3o = suc 2o | |
8 | oa1suc 7879 | . . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ (2o +o 1o) = suc 2o |
10 | 7, 9 | eqtr4i 2853 | . . . 4 ⊢ 3o = (2o +o 1o) |
11 | suceq 6029 | . . . 4 ⊢ (3o = (2o +o 1o) → suc 3o = suc (2o +o 1o)) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ suc 3o = suc (2o +o 1o) |
13 | 4, 6, 12 | 3eqtr4i 2860 | . 2 ⊢ (2o +o 2o) = suc 3o |
14 | df-4o 7830 | . 2 ⊢ 4o = suc 3o | |
15 | 13, 14 | eqtr4i 2853 | 1 ⊢ (2o +o 2o) = 4o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∈ wcel 2166 Oncon0 5964 suc csuc 5966 (class class class)co 6906 1oc1o 7820 2oc2o 7821 3oc3o 7822 4oc4o 7823 +o coa 7824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-2o 7828 df-3o 7829 df-4o 7830 df-oadd 7831 |
This theorem is referenced by: (None) |
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