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Theorem 1kp2ke3k 10278
Description: Example for df-dec 8428, 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 8428 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 8255 . . . 4  |-  1  e.  NN0
2 0nn0 8254 . . . 4  |-  0  e.  NN0
31, 2deccl 8441 . . 3  |- ; 1 0  e.  NN0
43, 2deccl 8441 . 2  |- ;; 1 0 0  e.  NN0
5 2nn0 8256 . . . 4  |-  2  e.  NN0
65, 2deccl 8441 . . 3  |- ; 2 0  e.  NN0
76, 2deccl 8441 . 2  |- ;; 2 0 0  e.  NN0
8 eqid 2056 . 2  |- ;;; 1 0 0 0  = ;;; 1 0 0 0
9 eqid 2056 . 2  |- ;;; 2 0 0 0  = ;;; 2 0 0 0
10 eqid 2056 . . 3  |- ;; 1 0 0  = ;; 1 0 0
11 eqid 2056 . . 3  |- ;; 2 0 0  = ;; 2 0 0
12 eqid 2056 . . . 4  |- ; 1 0  = ; 1 0
13 eqid 2056 . . . 4  |- ; 2 0  = ; 2 0
14 1p2e3 8117 . . . 4  |-  ( 1  +  2 )  =  3
15 00id 7215 . . . 4  |-  ( 0  +  0 )  =  0
161, 2, 5, 2, 12, 13, 14, 15decadd 8480 . . 3  |-  (; 1 0  + ; 2 0 )  = ; 3
0
173, 2, 6, 2, 10, 11, 16, 15decadd 8480 . 2  |-  (;; 1 0 0  + ;; 2 0 0 )  = ;; 3 0 0
184, 2, 7, 2, 8, 9, 17, 15decadd 8480 1  |-  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0
Colors of variables: wff set class
Syntax hints:    = wceq 1259  (class class class)co 5540   0cc0 6947   1c1 6948    + caddc 6950   2c2 8040   3c3 8041  ;cdc 8427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-inn 7991  df-2 8049  df-3 8050  df-4 8051  df-5 8052  df-6 8053  df-7 8054  df-8 8055  df-9 8056  df-n0 8240  df-dec 8428
This theorem is referenced by: (None)
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