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Mirrors > Home > ILE Home > Th. List > addsubass | Unicode version |
Description: Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addsubass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 992 | . . . . 5 | |
2 | subcl 8118 | . . . . . 6 | |
3 | 2 | 3adant1 1010 | . . . . 5 |
4 | simp3 994 | . . . . 5 | |
5 | 1, 3, 4 | addassd 7942 | . . . 4 |
6 | npcan 8128 | . . . . . 6 | |
7 | 6 | 3adant1 1010 | . . . . 5 |
8 | 7 | oveq2d 5869 | . . . 4 |
9 | 5, 8 | eqtrd 2203 | . . 3 |
10 | 9 | oveq1d 5868 | . 2 |
11 | 1, 3 | addcld 7939 | . . 3 |
12 | pncan 8125 | . . 3 | |
13 | 11, 4, 12 | syl2anc 409 | . 2 |
14 | 10, 13 | eqtr3d 2205 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 973 wceq 1348 wcel 2141 (class class class)co 5853 cc 7772 caddc 7777 cmin 8090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-resscn 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 |
This theorem is referenced by: addsub 8130 subadd23 8131 addsubeq4 8134 npncan 8140 subsub 8149 subsub3 8151 addsub4 8162 negsub 8167 addsubassi 8210 addsubassd 8250 zeo 9317 frecfzen2 10383 odd2np1 11832 |
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