unitadd - Mathbox for Stanislas Polu

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Theorem unitadd 44639

Description: Theorem used in conjunction with decaddc 12690 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.)

**Hypotheses**
Ref	Expression
unitadd.1	⊢ (𝐴 + 𝐵) = 𝐹
unitadd.2	⊢ (𝐶 + 1) = 𝐵
unitadd.3	⊢ 𝐴 ∈ ℕ₀
unitadd.4	⊢ 𝐶 ∈ ℕ₀

**Assertion**
Ref	Expression
unitadd	⊢ ((𝐴 + 𝐶) + 1) = 𝐹

**Proof of Theorem unitadd**
Step	Hyp	Ref	Expression
1		unitadd.3	. . . 4 ⊢ 𝐴 ∈ ℕ₀
2	1	nn0cni 12440	. . 3 ⊢ 𝐴 ∈ ℂ
3		unitadd.4	. . . 4 ⊢ 𝐶 ∈ ℕ₀
4	3	nn0cni 12440	. . 3 ⊢ 𝐶 ∈ ℂ
5		ax-1cn 11087	. . 3 ⊢ 1 ∈ ℂ
6	2, 4, 5	addassi 11146	. 2 ⊢ ((𝐴 + 𝐶) + 1) = (𝐴 + (𝐶 + 1))
7		unitadd.2	. . . . 5 ⊢ (𝐶 + 1) = 𝐵
8	7	eqcomi 2748	. . . 4 ⊢ 𝐵 = (𝐶 + 1)
9	8	oveq2i 7367	. . 3 ⊢ (𝐴 + 𝐵) = (𝐴 + (𝐶 + 1))
10		unitadd.1	. . 3 ⊢ (𝐴 + 𝐵) = 𝐹
11	9, 10	eqtr3i 2764	. 2 ⊢ (𝐴 + (𝐶 + 1)) = 𝐹
12	6, 11	eqtri 2762	1 ⊢ ((𝐴 + 𝐶) + 1) = 𝐹

Colors of variables: wff setvar class

Syntax hints: = wceq 1547 ∈ wcel 2119 (class class class)co 7356 1c1 11030 + caddc 11032 ℕ₀cn0 12428

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pr 5362 ax-un 7678 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-mulcl 11091 ax-addass 11094 ax-i2m1 11097

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-ov 7359 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-nn 12166 df-n0 12429

This theorem is referenced by: (None)

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