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Theorem 1kp2ke3k 20886
Description: Example for df-dec 10172, 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 10172 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  |-  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0

Proof of Theorem 1kp2ke3k
StepHypRef Expression
1 1nn0 10028 . . . 4  |-  1  e.  NN0
2 0nn0 10027 . . . 4  |-  0  e.  NN0
31, 2deccl 10185 . . 3  |- ; 1 0  e.  NN0
43, 2deccl 10185 . 2  |- ;; 1 0 0  e.  NN0
5 2nn0 10029 . . . 4  |-  2  e.  NN0
65, 2deccl 10185 . . 3  |- ; 2 0  e.  NN0
76, 2deccl 10185 . 2  |- ;; 2 0 0  e.  NN0
8 eqid 2316 . 2  |- ;;; 1 0 0 0  = ;;; 1 0 0 0
9 eqid 2316 . 2  |- ;;; 2 0 0 0  = ;;; 2 0 0 0
10 eqid 2316 . . 3  |- ;; 1 0 0  = ;; 1 0 0
11 eqid 2316 . . 3  |- ;; 2 0 0  = ;; 2 0 0
12 eqid 2316 . . . 4  |- ; 1 0  = ; 1 0
13 eqid 2316 . . . 4  |- ; 2 0  = ; 2 0
14 2cn 9861 . . . . 5  |-  2  e.  CC
15 ax-1cn 8840 . . . . 5  |-  1  e.  CC
16 2p1e3 9894 . . . . 5  |-  ( 2  +  1 )  =  3
1714, 15, 16addcomli 9049 . . . 4  |-  ( 1  +  2 )  =  3
18 00id 9032 . . . 4  |-  ( 0  +  0 )  =  0
191, 2, 5, 2, 12, 13, 17, 18decadd 10212 . . 3  |-  (; 1 0  + ; 2 0 )  = ; 3
0
203, 2, 6, 2, 10, 11, 19, 18decadd 10212 . 2  |-  (;; 1 0 0  + ;; 2 0 0 )  = ;; 3 0 0
214, 2, 7, 2, 8, 9, 20, 18decadd 10212 1  |-  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0
Colors of variables: wff set class
Syntax hints:    = wceq 1633  (class class class)co 5900   0cc0 8782   1c1 8783    + caddc 8785   2c2 9840   3c3 9841  ;cdc 10171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-ltxr 8917  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-dec 10172
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