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Mirrors > Home > ILE Home > Th. List > xpdjuen | Unicode version |
Description: Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
xpdjuen | ⊔ ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enrefg 6711 | . . . . . 6 | |
2 | 1 | 3ad2ant1 1003 | . . . . 5 |
3 | 0ex 4093 | . . . . . . 7 | |
4 | simp2 983 | . . . . . . 7 | |
5 | xpsnen2g 6776 | . . . . . . 7 | |
6 | 3, 4, 5 | sylancr 411 | . . . . . 6 |
7 | 6 | ensymd 6730 | . . . . 5 |
8 | xpen 6792 | . . . . 5 | |
9 | 2, 7, 8 | syl2anc 409 | . . . 4 |
10 | 1on 6372 | . . . . . . 7 | |
11 | simp3 984 | . . . . . . 7 | |
12 | xpsnen2g 6776 | . . . . . . 7 | |
13 | 10, 11, 12 | sylancr 411 | . . . . . 6 |
14 | 13 | ensymd 6730 | . . . . 5 |
15 | xpen 6792 | . . . . 5 | |
16 | 2, 14, 15 | syl2anc 409 | . . . 4 |
17 | xp01disjl 6383 | . . . . . . 7 | |
18 | 17 | xpeq2i 4609 | . . . . . 6 |
19 | xpindi 4723 | . . . . . 6 | |
20 | xp0 5007 | . . . . . 6 | |
21 | 18, 19, 20 | 3eqtr3i 2186 | . . . . 5 |
22 | 21 | a1i 9 | . . . 4 |
23 | djuenun 7149 | . . . 4 ⊔ | |
24 | 9, 16, 22, 23 | syl3anc 1220 | . . 3 ⊔ |
25 | df-dju 6984 | . . . . 5 ⊔ | |
26 | 25 | xpeq2i 4609 | . . . 4 ⊔ |
27 | xpundi 4644 | . . . 4 | |
28 | 26, 27 | eqtri 2178 | . . 3 ⊔ |
29 | 24, 28 | breqtrrdi 4008 | . 2 ⊔ ⊔ |
30 | 29 | ensymd 6730 | 1 ⊔ ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 963 wceq 1335 wcel 2128 cvv 2712 cun 3100 cin 3101 c0 3395 csn 3561 class class class wbr 3967 con0 4325 cxp 4586 c1o 6358 cen 6685 ⊔ cdju 6983 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-iord 4328 df-on 4330 df-suc 4333 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-1o 6365 df-er 6482 df-en 6688 df-dju 6984 df-inl 6993 df-inr 6994 |
This theorem is referenced by: (None) |
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