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Mirrors > Home > ILE Home > Th. List > strleund | Unicode version |
Description: Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
Ref | Expression |
---|---|
strleund.f | Struct |
strleund.g | Struct |
strleund.l |
Ref | Expression |
---|---|
strleund | Struct |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strleund.f | . . . . 5 Struct | |
2 | isstructim 12274 | . . . . 5 Struct | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | 3 | simp1d 994 | . . 3 |
5 | 4 | simp1d 994 | . 2 |
6 | strleund.g | . . . . 5 Struct | |
7 | isstructim 12274 | . . . . 5 Struct | |
8 | 6, 7 | syl 14 | . . . 4 |
9 | 8 | simp1d 994 | . . 3 |
10 | 9 | simp2d 995 | . 2 |
11 | 5 | nnred 8851 | . . 3 |
12 | 9 | simp1d 994 | . . . 4 |
13 | 12 | nnred 8851 | . . 3 |
14 | 10 | nnred 8851 | . . 3 |
15 | 4 | simp2d 995 | . . . . 5 |
16 | 15 | nnred 8851 | . . . 4 |
17 | 4 | simp3d 996 | . . . 4 |
18 | strleund.l | . . . . 5 | |
19 | 16, 13, 18 | ltled 7998 | . . . 4 |
20 | 11, 16, 13, 17, 19 | letrd 8003 | . . 3 |
21 | 9 | simp3d 996 | . . 3 |
22 | 11, 13, 14, 20, 21 | letrd 8003 | . 2 |
23 | 3 | simp2d 995 | . . . 4 |
24 | 8 | simp2d 995 | . . . 4 |
25 | difss 3234 | . . . . . . . 8 | |
26 | dmss 4787 | . . . . . . . 8 | |
27 | 25, 26 | mp1i 10 | . . . . . . 7 |
28 | 3 | simp3d 996 | . . . . . . 7 |
29 | 27, 28 | sstrd 3138 | . . . . . 6 |
30 | difss 3234 | . . . . . . . 8 | |
31 | dmss 4787 | . . . . . . . 8 | |
32 | 30, 31 | mp1i 10 | . . . . . . 7 |
33 | 8 | simp3d 996 | . . . . . . 7 |
34 | 32, 33 | sstrd 3138 | . . . . . 6 |
35 | ss2in 3336 | . . . . . 6 | |
36 | 29, 34, 35 | syl2anc 409 | . . . . 5 |
37 | fzdisj 9960 | . . . . . 6 | |
38 | 18, 37 | syl 14 | . . . . 5 |
39 | sseq0 3436 | . . . . 5 | |
40 | 36, 38, 39 | syl2anc 409 | . . . 4 |
41 | funun 5216 | . . . 4 | |
42 | 23, 24, 40, 41 | syl21anc 1219 | . . 3 |
43 | difundir 3361 | . . . 4 | |
44 | 43 | funeqi 5193 | . . 3 |
45 | 42, 44 | sylibr 133 | . 2 |
46 | structex 12272 | . . . 4 Struct | |
47 | 1, 46 | syl 14 | . . 3 |
48 | structex 12272 | . . . 4 Struct | |
49 | 6, 48 | syl 14 | . . 3 |
50 | unexg 4405 | . . 3 | |
51 | 47, 49, 50 | syl2anc 409 | . 2 |
52 | dmun 4795 | . . 3 | |
53 | 15 | nnzd 9290 | . . . . . . 7 |
54 | 10 | nnzd 9290 | . . . . . . 7 |
55 | 16, 13, 14, 19, 21 | letrd 8003 | . . . . . . 7 |
56 | eluz2 9450 | . . . . . . 7 | |
57 | 53, 54, 55, 56 | syl3anbrc 1166 | . . . . . 6 |
58 | fzss2 9972 | . . . . . 6 | |
59 | 57, 58 | syl 14 | . . . . 5 |
60 | 28, 59 | sstrd 3138 | . . . 4 |
61 | 5 | nnzd 9290 | . . . . . . 7 |
62 | 12 | nnzd 9290 | . . . . . . 7 |
63 | eluz2 9450 | . . . . . . 7 | |
64 | 61, 62, 20, 63 | syl3anbrc 1166 | . . . . . 6 |
65 | fzss1 9971 | . . . . . 6 | |
66 | 64, 65 | syl 14 | . . . . 5 |
67 | 33, 66 | sstrd 3138 | . . . 4 |
68 | 60, 67 | unssd 3284 | . . 3 |
69 | 52, 68 | eqsstrid 3174 | . 2 |
70 | isstructr 12275 | . 2 Struct | |
71 | 5, 10, 22, 45, 51, 69, 70 | syl33anc 1235 | 1 Struct |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 963 wceq 1335 wcel 2128 cvv 2712 cdif 3099 cun 3100 cin 3101 wss 3102 c0 3395 csn 3561 cop 3564 class class class wbr 3967 cdm 4588 wfun 5166 cfv 5172 (class class class)co 5826 clt 7914 cle 7915 cn 8838 cz 9172 cuz 9444 cfz 9918 Struct cstr 12256 |
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-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-addcom 7834 ax-addass 7836 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-ltadd 7850 |
This theorem depends on definitions: df-bi 116 df-3or 964 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-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 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-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 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-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-inn 8839 df-n0 9096 df-z 9173 df-uz 9445 df-fz 9919 df-struct 12262 |
This theorem is referenced by: strle2g 12351 strle3g 12352 srngstrd 12382 lmodstrd 12393 ipsstrd 12401 |
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