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Mirrors > Home > MPE Home > Th. List > addassi | Structured version Visualization version GIF version |
Description: Associative law for addition. (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
axi.2 | ⊢ 𝐵 ∈ ℂ |
axi.3 | ⊢ 𝐶 ∈ ℂ |
Ref | Expression |
---|---|
addassi | ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | axi.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | axi.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | addass 11267 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1461 | 1 ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 (class class class)co 7445 ℂcc 11178 + caddc 11183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-addass 11245 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 |
This theorem is referenced by: mul02lem2 11463 addrid 11466 2p2e4 12424 1p2e3 12432 3p2e5 12440 3p3e6 12441 4p2e6 12442 4p3e7 12443 4p4e8 12444 5p2e7 12445 5p3e8 12446 5p4e9 12447 6p2e8 12448 6p3e9 12449 7p2e9 12450 numsuc 12768 nummac 12799 numaddc 12802 6p5lem 12824 5p5e10 12825 6p4e10 12826 7p3e10 12829 8p2e10 12834 binom2i 14257 faclbnd4lem1 14338 3dvdsdec 16374 3dvds2dec 16375 gcdaddmlem 16564 mod2xnegi 17113 decexp2 17117 decsplit 17125 lgsdir2lem2 27379 2lgsoddprmlem3d 27466 ax5seglem7 28959 normlem3 31135 stadd3i 32271 dfdec100 32826 dp3mul10 32854 dpmul 32869 dpmul4 32870 quad3 35630 addassnni 41889 sn-1ne2 42202 sqmid3api 42220 re1m1e0m0 42306 sn-0tie0 42348 fltnltalem 42550 unitadd 44097 sqwvfoura 46083 sqwvfourb 46084 fouriersw 46086 3exp4mod41 47422 bgoldbtbndlem1 47611 |
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