dec0h - Intuitionistic Logic Explorer

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Theorem dec0h 9343

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

**Hypothesis**
Ref	Expression
dec0u.1	⊢ 𝐴 ∈ ℕ₀

**Assertion**
Ref	Expression
dec0h	⊢ 𝐴 = ;0𝐴

**Proof of Theorem dec0h**
Step	Hyp	Ref	Expression
1		10nn0 9339	. . 3 ⊢ ;10 ∈ ℕ₀
2		dec0u.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
3	1, 2	num0h 9333	. 2 ⊢ 𝐴 = ((;10 · 0) + 𝐴)
4		dfdec10 9325	. 2 ⊢ ;0𝐴 = ((;10 · 0) + 𝐴)
5	3, 4	eqtr4i 2189	1 ⊢ 𝐴 = ;0𝐴

Colors of variables: wff set class

Syntax hints: = wceq 1343 ∈ wcel 2136 (class class class)co 5842 0cc0 7753 1c1 7754 + caddc 7756 · cmul 7758 ℕ₀cn0 9114 cdc 9322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-sub 8071 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-5 8919 df-6 8920 df-7 8921 df-8 8922 df-9 8923 df-n0 9115 df-dec 9323

This theorem is referenced by: declei 9357 decrmanc 9378 decrmac 9379 decaddi 9381 decaddci 9382 decmulnc 9388