dec0h - Intuitionistic Logic Explorer

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Theorem dec0h 8568

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 8564	. . 3 ⊢ ;10 ∈ ℕ₀
2		dec0u.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
3	1, 2	num0h 8558	. 2 ⊢ 𝐴 = ((;10 · 0) + 𝐴)
4		dfdec10 8550	. 2 ⊢ ;0𝐴 = ((;10 · 0) + 𝐴)
5	3, 4	eqtr4i 2105	1 ⊢ 𝐴 = ;0𝐴

Colors of variables: wff set class

Syntax hints: = wceq 1285 ∈ wcel 1434 (class class class)co 5537 0cc0 7032 1c1 7033 + caddc 7035 · cmul 7037 ℕ₀cn0 8344 cdc 8547

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3898 ax-pow 3950 ax-pr 3966 ax-setind 4282 ax-cnex 7118 ax-resscn 7119 ax-1cn 7120 ax-1re 7121 ax-icn 7122 ax-addcl 7123 ax-addrcl 7124 ax-mulcl 7125 ax-addcom 7127 ax-mulcom 7128 ax-addass 7129 ax-mulass 7130 ax-distr 7131 ax-i2m1 7132 ax-1rid 7134 ax-0id 7135 ax-rnegex 7136 ax-cnre 7138

This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3386 df-sn 3406 df-pr 3407 df-op 3409 df-uni 3604 df-int 3639 df-br 3788 df-opab 3842 df-id 4050 df-xp 4371 df-rel 4372 df-cnv 4373 df-co 4374 df-dm 4375 df-iota 4891 df-fun 4928 df-fv 4934 df-riota 5493 df-ov 5540 df-oprab 5541 df-mpt2 5542 df-sub 7337 df-inn 8096 df-2 8154 df-3 8155 df-4 8156 df-5 8157 df-6 8158 df-7 8159 df-8 8160 df-9 8161 df-n0 8345 df-dec 8548

This theorem is referenced by: declei 8582 decrmanc 8603 decrmac 8604 decaddi 8606 decaddci 8607 decmulnc 8613