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Theorem 1kp2ke3k 26056
Description: Example for df-dec 11142, 1000 + 2000 = 3000.

This proof disproves (by counter-example) 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 11142 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 10976 . . . 4 1 ∈ ℕ0
2 0nn0 10975 . . . 4 0 ∈ ℕ0
31, 2deccl 11155 . . 3 10 ∈ ℕ0
43, 2deccl 11155 . 2 100 ∈ ℕ0
5 2nn0 10977 . . . 4 2 ∈ ℕ0
65, 2deccl 11155 . . 3 20 ∈ ℕ0
76, 2deccl 11155 . 2 200 ∈ ℕ0
8 eqid 2505 . 2 1000 = 1000
9 eqid 2505 . 2 2000 = 2000
10 eqid 2505 . . 3 100 = 100
11 eqid 2505 . . 3 200 = 200
12 eqid 2505 . . . 4 10 = 10
13 eqid 2505 . . . 4 20 = 20
14 1p2e3 10824 . . . 4 (1 + 2) = 3
15 00id 9893 . . . 4 (0 + 0) = 0
161, 2, 5, 2, 12, 13, 14, 15decadd 11182 . . 3 (10 + 20) = 30
173, 2, 6, 2, 10, 11, 16, 15decadd 11182 . 2 (100 + 200) = 300
184, 2, 7, 2, 8, 9, 17, 15decadd 11182 1 (1000 + 2000) = 3000
Colors of variables: wff setvar class
Syntax hints:   = wceq 1468  (class class class)co 6363  0cc0 9624  1c1 9625   + caddc 9627  2c2 10748  3c3 10749  cdc 11141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659  ax-resscn 9681  ax-1cn 9682  ax-icn 9683  ax-addcl 9684  ax-addrcl 9685  ax-mulcl 9686  ax-mulrcl 9687  ax-mulcom 9688  ax-addass 9689  ax-mulass 9690  ax-distr 9691  ax-i2m1 9692  ax-1ne0 9693  ax-1rid 9694  ax-rnegex 9695  ax-rrecex 9696  ax-cnre 9697  ax-pre-lttri 9698  ax-pre-lttrn 9699  ax-pre-ltadd 9700
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-nel 2678  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-ov 6366  df-om 6770  df-wrecs 7105  df-recs 7167  df-rdg 7205  df-er 7440  df-en 7653  df-dom 7654  df-sdom 7655  df-pnf 9762  df-mnf 9763  df-ltxr 9765  df-nn 10699  df-2 10757  df-3 10758  df-4 10759  df-5 10760  df-6 10761  df-7 10762  df-8 10763  df-9 10764  df-10 10765  df-n0 10961  df-dec 11142
This theorem is referenced by: (None)
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