11multnc - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 11multnc		GIF version

Theorem 11multnc 9827

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

**Hypothesis**
Ref	Expression
11multnc.n	⊢ 𝑁 ∈ ℕ₀

**Assertion**
Ref	Expression
11multnc	⊢ (𝑁 · ;11) = ;𝑁𝑁

**Proof of Theorem 11multnc**
Step	Hyp	Ref	Expression
1		11multnc.n	. . 3 ⊢ 𝑁 ∈ ℕ₀
2		1nn0 9562	. . 3 ⊢ 1 ∈ ℕ₀
3	1, 2, 2	decmulnc 9826	. 2 ⊢ (𝑁 · ;11) = ;(𝑁 · 1)(𝑁 · 1)
4	1	nn0cni 9558	. . . 4 ⊢ 𝑁 ∈ ℂ
5	4	mulridi 8322	. . 3 ⊢ (𝑁 · 1) = 𝑁
6	5, 5	deceq12i 9768	. 2 ⊢ ;(𝑁 · 1)(𝑁 · 1) = ;𝑁𝑁
7	3, 6	eqtri 2259	1 ⊢ (𝑁 · ;11) = ;𝑁𝑁

Colors of variables: wff set class

Syntax hints: = wceq 1402 ∈ wcel 2209 (class class class)co 6079 1c1 8174 · cmul 8178 ℕ₀cn0 9546 cdc 9760

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-pr 4344 ax-setind 4682 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-cnre 8284

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-br 4129 df-opab 4191 df-id 4436 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-iota 5335 df-fun 5377 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-sub 8493 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-5 9349 df-6 9350 df-7 9351 df-8 9352 df-9 9353 df-n0 9547 df-dec 9761

This theorem is referenced by: birthdaylog2 16073