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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdeceq2i 9201 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq12i 9202 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremnumnncl 9203 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0u 9204 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0h 9205 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumcl 9206 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumsuc 9207 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeccl 9208 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem10nn 9209 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
;

Theorem10pos 9210 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theorem10nn0 9211 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem10re 9212 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theoremdecnncl 9213 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theoremdec0u 9214 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdec0h 9215 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theoremnumnncl2 9216 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremdecnncl2 9217 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theoremnumlt 9218 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumltc 9219 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremle9lt10 9220 A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.)
;

Theoremdeclt 9221 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdecltc 9222 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;              ; ;

Theoremdeclth 9223 Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
; ;

Theoremdecsuc 9224 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;       ;

Theorem3declth 9225 Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
;; ;;

Theorem3decltc 9226 Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
;       ;       ;; ;;

Theoremdecle 9227 Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theoremdecleh 9228 Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theoremdeclei 9229 Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
;

Theoremnumlti 9230 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeclti 9231 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;

Theoremdecltdi 9232 Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
;

Theoremnumsucc 9233 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecsucc 9234 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;

Theorem1e0p1 9235 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdec10p 9236 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremnumma 9237 Perform a multiply-add of two decimal integers and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummac 9238 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumma2c 9239 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumadd 9240 Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumaddc 9241 Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul1c 9242 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul2c 9243 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecma 9244 Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                            ;

Theoremdecmac 9245 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                                   ;       ;

Theoremdecma2c 9246 Perform a multiply-add of two numerals and against a fixed multiplier (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                                   ;       ;

Theoremdecadd 9247 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                     ;

Theoremdecaddc 9248 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                     ;       ;

Theoremdecaddc2 9249 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;              ;       ;

Theoremdecrmanc 9250 Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by AV, 16-Sep-2021.)
;                            ;

Theoremdecrmac 9251 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by AV, 16-Sep-2021.)
;                                   ;       ;

Theoremdecaddm10 9252 The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
; ; ;

Theoremdecaddi 9253 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;              ;

Theoremdecaddci 9254 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;       ;

Theoremdecaddci2 9255 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;              ;       ;

Theoremdecsubi 9256 Difference between a numeral and a nonnegative integer (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
;                     ;

Theoremdecmul1 9257 The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
;                            ;

Theoremdecmul1c 9258 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;                            ;       ;

Theoremdecmul2c 9259 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;                            ;       ;

Theoremdecmulnc 9260 The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem11multnc 9261 The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theoremdecmul10add 9262 A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem6p5lem 9263 Lemma for 6p5e11 9266 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
;       ;

Theorem5p5e10 9264 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem6p4e10 9265 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem6p5e11 9266 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem6p6e12 9267 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p3e10 9268 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem7p4e11 9269 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem7p5e12 9270 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p6e13 9271 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p7e14 9272 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p2e10 9273 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem8p3e11 9274 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem8p4e12 9275 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p5e13 9276 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p6e14 9277 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p7e15 9278 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p8e16 9279 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p2e11 9280 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem9p3e12 9281 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p4e13 9282 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p5e14 9283 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p6e15 9284 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p7e16 9285 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p8e17 9286 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p9e18 9287 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem10p10e20 9288 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem10m1e9 9289 10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
;

Theorem4t3lem 9290 Lemma for 4t3e12 9291 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem4t3e12 9291 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem4t4e16 9292 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem5t2e10 9293 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
;

Theorem5t3e15 9294 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem5t4e20 9295 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem5t5e25 9296 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem6t2e12 9297 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t3e18 9298 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t4e24 9299 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t5e30 9300 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

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