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Theorem 1kp2ke3k 26437
Description: Example for df-dec 11238, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with (2 + 1) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 11238 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

Assertion
Ref Expression
1kp2ke3k (1000 + 2000) = 3000

Proof of Theorem 1kp2ke3k
StepHypRef Expression
1 1nn0 11067 . . . 4 1 ∈ ℕ0
2 0nn0 11066 . . . 4 0 ∈ ℕ0
31, 2deccl 11256 . . 3 10 ∈ ℕ0
43, 2deccl 11256 . 2 100 ∈ ℕ0
5 2nn0 11068 . . . 4 2 ∈ ℕ0
65, 2deccl 11256 . . 3 20 ∈ ℕ0
76, 2deccl 11256 . 2 200 ∈ ℕ0
8 eqid 2514 . 2 1000 = 1000
9 eqid 2514 . 2 2000 = 2000
10 eqid 2514 . . 3 100 = 100
11 eqid 2514 . . 3 200 = 200
12 eqid 2514 . . . 4 10 = 10
13 eqid 2514 . . . 4 20 = 20
14 1p2e3 10911 . . . 4 (1 + 2) = 3
15 00id 9966 . . . 4 (0 + 0) = 0
161, 2, 5, 2, 12, 13, 14, 15decadd 11314 . . 3 (10 + 20) = 30
173, 2, 6, 2, 10, 11, 16, 15decadd 11314 . 2 (100 + 200) = 300
184, 2, 7, 2, 8, 9, 17, 15decadd 11314 1 (1000 + 2000) = 3000
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  (class class class)co 6431  0cc0 9695  1c1 9696   + caddc 9698  2c2 10829  3c3 10830  cdc 11237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6728  ax-resscn 9752  ax-1cn 9753  ax-icn 9754  ax-addcl 9755  ax-addrcl 9756  ax-mulcl 9757  ax-mulrcl 9758  ax-mulcom 9759  ax-addass 9760  ax-mulass 9761  ax-distr 9762  ax-i2m1 9763  ax-1ne0 9764  ax-1rid 9765  ax-rnegex 9766  ax-rrecex 9767  ax-cnre 9768  ax-pre-lttri 9769  ax-pre-lttrn 9770  ax-pre-ltadd 9771
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5658  df-fun 5696  df-fn 5697  df-f 5698  df-f1 5699  df-fo 5700  df-f1o 5701  df-fv 5702  df-ov 6434  df-om 6839  df-wrecs 7174  df-recs 7235  df-rdg 7273  df-er 7509  df-en 7722  df-dom 7723  df-sdom 7724  df-pnf 9835  df-mnf 9836  df-ltxr 9838  df-nn 10780  df-2 10838  df-3 10839  df-4 10840  df-5 10841  df-6 10842  df-7 10843  df-8 10844  df-9 10845  df-n0 11052  df-dec 11238
This theorem is referenced by: (None)
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