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Theorem 1kp2ke3k 21746
Description: Example for df-dec 10375, 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 10375 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 10229 . . . 4  |-  1  e.  NN0
2 0nn0 10228 . . . 4  |-  0  e.  NN0
31, 2deccl 10388 . . 3  |- ; 1 0  e.  NN0
43, 2deccl 10388 . 2  |- ;; 1 0 0  e.  NN0
5 2nn0 10230 . . . 4  |-  2  e.  NN0
65, 2deccl 10388 . . 3  |- ; 2 0  e.  NN0
76, 2deccl 10388 . 2  |- ;; 2 0 0  e.  NN0
8 eqid 2435 . 2  |- ;;; 1 0 0 0  = ;;; 1 0 0 0
9 eqid 2435 . 2  |- ;;; 2 0 0 0  = ;;; 2 0 0 0
10 eqid 2435 . . 3  |- ;; 1 0 0  = ;; 1 0 0
11 eqid 2435 . . 3  |- ;; 2 0 0  = ;; 2 0 0
12 eqid 2435 . . . 4  |- ; 1 0  = ; 1 0
13 eqid 2435 . . . 4  |- ; 2 0  = ; 2 0
14 2cn 10062 . . . . 5  |-  2  e.  CC
15 ax-1cn 9040 . . . . 5  |-  1  e.  CC
16 2p1e3 10095 . . . . 5  |-  ( 2  +  1 )  =  3
1714, 15, 16addcomli 9250 . . . 4  |-  ( 1  +  2 )  =  3
18 00id 9233 . . . 4  |-  ( 0  +  0 )  =  0
191, 2, 5, 2, 12, 13, 17, 18decadd 10415 . . 3  |-  (; 1 0  + ; 2 0 )  = ; 3
0
203, 2, 6, 2, 10, 11, 19, 18decadd 10415 . 2  |-  (;; 1 0 0  + ;; 2 0 0 )  = ;; 3 0 0
214, 2, 7, 2, 8, 9, 20, 18decadd 10415 1  |-  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0
Colors of variables: wff set class
Syntax hints:    = wceq 1652  (class class class)co 6073   0cc0 8982   1c1 8983    + caddc 8985   2c2 10041   3c3 10042  ;cdc 10374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-dec 10375
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