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Mirrors > Home > ILE Home > Th. List > 1kp2ke3k | GIF version |
Description: Example for df-dec 9416, 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 9416 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 9223 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 0nn0 9222 | . . . 4 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 9429 | . . 3 ⊢ ;10 ∈ ℕ0 |
4 | 3, 2 | deccl 9429 | . 2 ⊢ ;;100 ∈ ℕ0 |
5 | 2nn0 9224 | . . . 4 ⊢ 2 ∈ ℕ0 | |
6 | 5, 2 | deccl 9429 | . . 3 ⊢ ;20 ∈ ℕ0 |
7 | 6, 2 | deccl 9429 | . 2 ⊢ ;;200 ∈ ℕ0 |
8 | eqid 2189 | . 2 ⊢ ;;;1000 = ;;;1000 | |
9 | eqid 2189 | . 2 ⊢ ;;;2000 = ;;;2000 | |
10 | eqid 2189 | . . 3 ⊢ ;;100 = ;;100 | |
11 | eqid 2189 | . . 3 ⊢ ;;200 = ;;200 | |
12 | eqid 2189 | . . . 4 ⊢ ;10 = ;10 | |
13 | eqid 2189 | . . . 4 ⊢ ;20 = ;20 | |
14 | 1p2e3 9084 | . . . 4 ⊢ (1 + 2) = 3 | |
15 | 00id 8129 | . . . 4 ⊢ (0 + 0) = 0 | |
16 | 1, 2, 5, 2, 12, 13, 14, 15 | decadd 9468 | . . 3 ⊢ (;10 + ;20) = ;30 |
17 | 3, 2, 6, 2, 10, 11, 16, 15 | decadd 9468 | . 2 ⊢ (;;100 + ;;200) = ;;300 |
18 | 4, 2, 7, 2, 8, 9, 17, 15 | decadd 9468 | 1 ⊢ (;;;1000 + ;;;2000) = ;;;3000 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 (class class class)co 5897 0cc0 7842 1c1 7843 + caddc 7845 2c2 9001 3c3 9002 ;cdc 9415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-sub 8161 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-5 9012 df-6 9013 df-7 9014 df-8 9015 df-9 9016 df-n0 9208 df-dec 9416 |
This theorem is referenced by: (None) |
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