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Mirrors > Home > MPE Home > Th. List > o2p2e4OLD | Structured version Visualization version GIF version |
Description: Obsolete version of o2p2e4 8491 as of 23-Mar-2024. (Contributed by NM, 18-Aug-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
o2p2e4OLD | ⊢ (2o +o 2o) = 4o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on 8430 | . . . 4 ⊢ 2o ∈ On | |
2 | 1on 8428 | . . . 4 ⊢ 1o ∈ On | |
3 | oasuc 8474 | . . . 4 ⊢ ((2o ∈ On ∧ 1o ∈ On) → (2o +o suc 1o) = suc (2o +o 1o)) | |
4 | 1, 2, 3 | mp2an 691 | . . 3 ⊢ (2o +o suc 1o) = suc (2o +o 1o) |
5 | df-2o 8417 | . . . 4 ⊢ 2o = suc 1o | |
6 | 5 | oveq2i 7372 | . . 3 ⊢ (2o +o 2o) = (2o +o suc 1o) |
7 | df-3o 8418 | . . . . 5 ⊢ 3o = suc 2o | |
8 | oa1suc 8481 | . . . . . 6 ⊢ (2o ∈ On → (2o +o 1o) = suc 2o) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ (2o +o 1o) = suc 2o |
10 | 7, 9 | eqtr4i 2764 | . . . 4 ⊢ 3o = (2o +o 1o) |
11 | suceq 6387 | . . . 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 2771 | . 2 ⊢ (2o +o 2o) = suc 3o |
14 | df-4o 8419 | . 2 ⊢ 4o = suc 3o | |
15 | 13, 14 | eqtr4i 2764 | 1 ⊢ (2o +o 2o) = 4o |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Oncon0 6321 suc csuc 6323 (class class class)co 7361 1oc1o 8409 2oc2o 8410 3oc3o 8411 4oc4o 8412 +o coa 8413 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-3o 8418 df-4o 8419 df-oadd 8420 |
This theorem is referenced by: (None) |
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