11multnc - Intuitionistic Logic Explorer

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Theorem 11multnc 9775

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 9511	. . 3 ⊢ 1 ∈ ℕ₀
3	1, 2, 2	decmulnc 9774	. 2 ⊢ (𝑁 · ;11) = ;(𝑁 · 1)(𝑁 · 1)
4	1	nn0cni 9507	. . . 4 ⊢ 𝑁 ∈ ℂ
5	4	mulridi 8275	. . 3 ⊢ (𝑁 · 1) = 𝑁
6	5, 5	deceq12i 9716	. 2 ⊢ ;(𝑁 · 1)(𝑁 · 1) = ;𝑁𝑁
7	3, 6	eqtri 2253	1 ⊢ (𝑁 · ;11) = ;𝑁𝑁

Colors of variables: wff set class

Syntax hints: = wceq 1398 ∈ wcel 2203 (class class class)co 6049 1c1 8127 · cmul 8131 ℕ₀cn0 9495 cdc 9708

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2206 ax-ext 2214 ax-sep 4227 ax-pow 4286 ax-pr 4321 ax-setind 4658 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-cnre 8237

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2814 df-sbc 3042 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-br 4109 df-opab 4171 df-id 4413 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-iota 5311 df-fun 5353 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-sub 8445 df-inn 9237 df-2 9295 df-3 9296 df-4 9297 df-5 9298 df-6 9299 df-7 9300 df-8 9301 df-9 9302 df-n0 9496 df-dec 9709

This theorem is referenced by: (None)