unitadd - Mathbox for Stanislas Polu

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Theorem unitadd 44185

Description: Theorem used in conjunction with decaddc 12786 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 12536	. . 3 ⊢ 𝐴 ∈ ℂ
3		unitadd.4	. . . 4 ⊢ 𝐶 ∈ ℕ₀
4	3	nn0cni 12536	. . 3 ⊢ 𝐶 ∈ ℂ
5		ax-1cn 11211	. . 3 ⊢ 1 ∈ ℂ
6	2, 4, 5	addassi 11269	. 2 ⊢ ((𝐴 + 𝐶) + 1) = (𝐴 + (𝐶 + 1))
7		unitadd.2	. . . . 5 ⊢ (𝐶 + 1) = 𝐵
8	7	eqcomi 2744	. . . 4 ⊢ 𝐵 = (𝐶 + 1)
9	8	oveq2i 7442	. . 3 ⊢ (𝐴 + 𝐵) = (𝐴 + (𝐶 + 1))
10		unitadd.1	. . 3 ⊢ (𝐴 + 𝐵) = 𝐹
11	9, 10	eqtr3i 2765	. 2 ⊢ (𝐴 + (𝐶 + 1)) = 𝐹
12	6, 11	eqtri 2763	1 ⊢ ((𝐴 + 𝐶) + 1) = 𝐹

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 1c1 11154 + caddc 11156 ℕ₀cn0 12524

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-addass 11218 ax-i2m1 11221

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-n0 12525

This theorem is referenced by: (None)

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