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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnaddcomli | Structured version Visualization version GIF version |
Description: Version of addcomli 10825 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.) |
Ref | Expression |
---|---|
nnaddcomli.1 | ⊢ 𝐴 ∈ ℕ |
nnaddcomli.2 | ⊢ 𝐵 ∈ ℕ |
nnaddcomli.3 | ⊢ (𝐴 + 𝐵) = 𝐶 |
Ref | Expression |
---|---|
nnaddcomli | ⊢ (𝐵 + 𝐴) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnaddcomli.2 | . . 3 ⊢ 𝐵 ∈ ℕ | |
2 | nnaddcomli.1 | . . 3 ⊢ 𝐴 ∈ ℕ | |
3 | nnaddcom 39238 | . . 3 ⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (𝐵 + 𝐴) = (𝐴 + 𝐵)) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (𝐵 + 𝐴) = (𝐴 + 𝐵) |
5 | nnaddcomli.3 | . 2 ⊢ (𝐴 + 𝐵) = 𝐶 | |
6 | 4, 5 | eqtri 2843 | 1 ⊢ (𝐵 + 𝐴) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 (class class class)co 7149 + caddc 10533 ℕcn 11631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-1cn 10588 ax-addcl 10590 ax-addass 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-nn 11632 |
This theorem is referenced by: (None) |
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