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Mirrors > Home > MPE Home > Th. List > 1kp2ke3k | Structured version Visualization version GIF version |
Description: Example for df-dec 11370, 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 11370 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 11185 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 0nn0 11184 | . . . 4 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 11388 | . . 3 ⊢ ;10 ∈ ℕ0 |
4 | 3, 2 | deccl 11388 | . 2 ⊢ ;;100 ∈ ℕ0 |
5 | 2nn0 11186 | . . . 4 ⊢ 2 ∈ ℕ0 | |
6 | 5, 2 | deccl 11388 | . . 3 ⊢ ;20 ∈ ℕ0 |
7 | 6, 2 | deccl 11388 | . 2 ⊢ ;;200 ∈ ℕ0 |
8 | eqid 2610 | . 2 ⊢ ;;;1000 = ;;;1000 | |
9 | eqid 2610 | . 2 ⊢ ;;;2000 = ;;;2000 | |
10 | eqid 2610 | . . 3 ⊢ ;;100 = ;;100 | |
11 | eqid 2610 | . . 3 ⊢ ;;200 = ;;200 | |
12 | eqid 2610 | . . . 4 ⊢ ;10 = ;10 | |
13 | eqid 2610 | . . . 4 ⊢ ;20 = ;20 | |
14 | 1p2e3 11029 | . . . 4 ⊢ (1 + 2) = 3 | |
15 | 00id 10090 | . . . 4 ⊢ (0 + 0) = 0 | |
16 | 1, 2, 5, 2, 12, 13, 14, 15 | decadd 11446 | . . 3 ⊢ (;10 + ;20) = ;30 |
17 | 3, 2, 6, 2, 10, 11, 16, 15 | decadd 11446 | . 2 ⊢ (;;100 + ;;200) = ;;300 |
18 | 4, 2, 7, 2, 8, 9, 17, 15 | decadd 11446 | 1 ⊢ (;;;1000 + ;;;2000) = ;;;3000 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 2c2 10947 3c3 10948 ;cdc 11369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-dec 11370 |
This theorem is referenced by: (None) |
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