11multnc - Intuitionistic Logic Explorer

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Theorem 11multnc 9007

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 8752	. . 3 ⊢ 1 ∈ ℕ₀
3	1, 2, 2	decmulnc 9006	. 2 ⊢ (𝑁 · ;11) = ;(𝑁 · 1)(𝑁 · 1)
4	1	nn0cni 8748	. . . 4 ⊢ 𝑁 ∈ ℂ
5	4	mulid1i 7553	. . 3 ⊢ (𝑁 · 1) = 𝑁
6	5, 5	deceq12i 8948	. 2 ⊢ ;(𝑁 · 1)(𝑁 · 1) = ;𝑁𝑁
7	3, 6	eqtri 2109	1 ⊢ (𝑁 · ;11) = ;𝑁𝑁

Colors of variables: wff set class

Syntax hints: = wceq 1290 ∈ wcel 1439 (class class class)co 5668 1c1 7414 · cmul 7418 ℕ₀cn0 8736 cdc 8940

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519

This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-br 3854 df-opab 3908 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-iota 4995 df-fun 5032 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-sub 7718 df-inn 8486 df-2 8544 df-3 8545 df-4 8546 df-5 8547 df-6 8548 df-7 8549 df-8 8550 df-9 8551 df-n0 8737 df-dec 8941

This theorem is referenced by: (None)