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Mirrors > Home > ILE Home > Th. List > 1kp2ke3k | GIF version |
Description: Example for df-dec 9183, 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 9183 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.) |
Ref | Expression |
---|---|
1kp2ke3k | ⊢ (;;;1000 + ;;;2000) = ;;;3000 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 8993 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 0nn0 8992 | . . . 4 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 9196 | . . 3 ⊢ ;10 ∈ ℕ0 |
4 | 3, 2 | deccl 9196 | . 2 ⊢ ;;100 ∈ ℕ0 |
5 | 2nn0 8994 | . . . 4 ⊢ 2 ∈ ℕ0 | |
6 | 5, 2 | deccl 9196 | . . 3 ⊢ ;20 ∈ ℕ0 |
7 | 6, 2 | deccl 9196 | . 2 ⊢ ;;200 ∈ ℕ0 |
8 | eqid 2139 | . 2 ⊢ ;;;1000 = ;;;1000 | |
9 | eqid 2139 | . 2 ⊢ ;;;2000 = ;;;2000 | |
10 | eqid 2139 | . . 3 ⊢ ;;100 = ;;100 | |
11 | eqid 2139 | . . 3 ⊢ ;;200 = ;;200 | |
12 | eqid 2139 | . . . 4 ⊢ ;10 = ;10 | |
13 | eqid 2139 | . . . 4 ⊢ ;20 = ;20 | |
14 | 1p2e3 8854 | . . . 4 ⊢ (1 + 2) = 3 | |
15 | 00id 7903 | . . . 4 ⊢ (0 + 0) = 0 | |
16 | 1, 2, 5, 2, 12, 13, 14, 15 | decadd 9235 | . . 3 ⊢ (;10 + ;20) = ;30 |
17 | 3, 2, 6, 2, 10, 11, 16, 15 | decadd 9235 | . 2 ⊢ (;;100 + ;;200) = ;;300 |
18 | 4, 2, 7, 2, 8, 9, 17, 15 | decadd 9235 | 1 ⊢ (;;;1000 + ;;;2000) = ;;;3000 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 (class class class)co 5774 0cc0 7620 1c1 7621 + caddc 7623 2c2 8771 3c3 8772 ;cdc 9182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 df-n0 8978 df-dec 9183 |
This theorem is referenced by: (None) |
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