11multnc - Intuitionistic Logic Explorer

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Theorem 11multnc 9584

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 9324	. . 3 ⊢ 1 ∈ ℕ₀
3	1, 2, 2	decmulnc 9583	. 2 ⊢ (𝑁 · ;11) = ;(𝑁 · 1)(𝑁 · 1)
4	1	nn0cni 9320	. . . 4 ⊢ 𝑁 ∈ ℂ
5	4	mulridi 8087	. . 3 ⊢ (𝑁 · 1) = 𝑁
6	5, 5	deceq12i 9525	. 2 ⊢ ;(𝑁 · 1)(𝑁 · 1) = ;𝑁𝑁
7	3, 6	eqtri 2227	1 ⊢ (𝑁 · ;11) = ;𝑁𝑁

Colors of variables: wff set class

Syntax hints: = wceq 1373 ∈ wcel 2177 (class class class)co 5954 1c1 7939 · cmul 7943 ℕ₀cn0 9308 cdc 9517

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-14 2180 ax-ext 2188 ax-sep 4167 ax-pow 4223 ax-pr 4258 ax-setind 4590 ax-cnex 8029 ax-resscn 8030 ax-1cn 8031 ax-1re 8032 ax-icn 8033 ax-addcl 8034 ax-addrcl 8035 ax-mulcl 8036 ax-addcom 8038 ax-mulcom 8039 ax-addass 8040 ax-mulass 8041 ax-distr 8042 ax-i2m1 8043 ax-1rid 8045 ax-0id 8046 ax-rnegex 8047 ax-cnre 8049

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3001 df-dif 3170 df-un 3172 df-in 3174 df-ss 3181 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-int 3889 df-br 4049 df-opab 4111 df-id 4345 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-iota 5238 df-fun 5279 df-fv 5285 df-riota 5909 df-ov 5957 df-oprab 5958 df-mpo 5959 df-sub 8258 df-inn 9050 df-2 9108 df-3 9109 df-4 9110 df-5 9111 df-6 9112 df-7 9113 df-8 9114 df-9 9115 df-n0 9309 df-dec 9518

This theorem is referenced by: (None)