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Mirrors > Home > ILE Home > Th. List > addsubeq4 | Unicode version |
Description: Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Ref | Expression |
---|---|
addsubeq4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2166 | . . 3 | |
2 | subcl 8089 | . . . . . 6 | |
3 | 2 | ancoms 266 | . . . . 5 |
4 | subadd 8093 | . . . . . . 7 | |
5 | 4 | 3expa 1192 | . . . . . 6 |
6 | 5 | ancoms 266 | . . . . 5 |
7 | 3, 6 | sylan 281 | . . . 4 |
8 | 7 | an4s 578 | . . 3 |
9 | 1, 8 | syl5bb 191 | . 2 |
10 | addcom 8027 | . . . . . . 7 | |
11 | 10 | adantl 275 | . . . . . 6 |
12 | 11 | oveq1d 5852 | . . . . 5 |
13 | addsubass 8100 | . . . . . . . 8 | |
14 | 13 | 3com12 1196 | . . . . . . 7 |
15 | 14 | 3expa 1192 | . . . . . 6 |
16 | 15 | ancoms 266 | . . . . 5 |
17 | 12, 16 | eqtrd 2197 | . . . 4 |
18 | 17 | adantlr 469 | . . 3 |
19 | 18 | eqeq1d 2173 | . 2 |
20 | addcl 7870 | . . 3 | |
21 | subadd 8093 | . . . . 5 | |
22 | 21 | 3expb 1193 | . . . 4 |
23 | 22 | ancoms 266 | . . 3 |
24 | 20, 23 | sylan2 284 | . 2 |
25 | 9, 19, 24 | 3bitr2rd 216 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 (class class class)co 5837 cc 7743 caddc 7748 cmin 8061 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-setind 4509 ax-resscn 7837 ax-1cn 7838 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-iota 5148 df-fun 5185 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-sub 8063 |
This theorem is referenced by: subcan 8145 addsubeq4d 8252 |
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