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Mirrors > Home > MPE Home > Th. List > i1fadd | Structured version Visualization version Unicode version |
Description: The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
i1fadd.1 | |
i1fadd.2 |
Ref | Expression |
---|---|
i1fadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | readdcl 10019 | . . . 4 | |
2 | 1 | adantl 482 | . . 3 |
3 | i1fadd.1 | . . . 4 | |
4 | i1ff 23443 | . . . 4 | |
5 | 3, 4 | syl 17 | . . 3 |
6 | i1fadd.2 | . . . 4 | |
7 | i1ff 23443 | . . . 4 | |
8 | 6, 7 | syl 17 | . . 3 |
9 | reex 10027 | . . . 4 | |
10 | 9 | a1i 11 | . . 3 |
11 | inidm 3822 | . . 3 | |
12 | 2, 5, 8, 10, 10, 11 | off 6912 | . 2 |
13 | i1frn 23444 | . . . . . 6 | |
14 | 3, 13 | syl 17 | . . . . 5 |
15 | i1frn 23444 | . . . . . 6 | |
16 | 6, 15 | syl 17 | . . . . 5 |
17 | xpfi 8231 | . . . . 5 | |
18 | 14, 16, 17 | syl2anc 693 | . . . 4 |
19 | eqid 2622 | . . . . . 6 | |
20 | ovex 6678 | . . . . . 6 | |
21 | 19, 20 | fnmpt2i 7239 | . . . . 5 |
22 | dffn4 6121 | . . . . 5 | |
23 | 21, 22 | mpbi 220 | . . . 4 |
24 | fofi 8252 | . . . 4 | |
25 | 18, 23, 24 | sylancl 694 | . . 3 |
26 | eqid 2622 | . . . . . . . . 9 | |
27 | rspceov 6692 | . . . . . . . . 9 | |
28 | 26, 27 | mp3an3 1413 | . . . . . . . 8 |
29 | ovex 6678 | . . . . . . . . 9 | |
30 | eqeq1 2626 | . . . . . . . . . 10 | |
31 | 30 | 2rexbidv 3057 | . . . . . . . . 9 |
32 | 29, 31 | elab 3350 | . . . . . . . 8 |
33 | 28, 32 | sylibr 224 | . . . . . . 7 |
34 | 33 | adantl 482 | . . . . . 6 |
35 | ffn 6045 | . . . . . . . 8 | |
36 | 5, 35 | syl 17 | . . . . . . 7 |
37 | dffn3 6054 | . . . . . . 7 | |
38 | 36, 37 | sylib 208 | . . . . . 6 |
39 | ffn 6045 | . . . . . . . 8 | |
40 | 8, 39 | syl 17 | . . . . . . 7 |
41 | dffn3 6054 | . . . . . . 7 | |
42 | 40, 41 | sylib 208 | . . . . . 6 |
43 | 34, 38, 42, 10, 10, 11 | off 6912 | . . . . 5 |
44 | frn 6053 | . . . . 5 | |
45 | 43, 44 | syl 17 | . . . 4 |
46 | 19 | rnmpt2 6770 | . . . 4 |
47 | 45, 46 | syl6sseqr 3652 | . . 3 |
48 | ssfi 8180 | . . 3 | |
49 | 25, 47, 48 | syl2anc 693 | . 2 |
50 | frn 6053 | . . . . . . . 8 | |
51 | 12, 50 | syl 17 | . . . . . . 7 |
52 | 51 | ssdifssd 3748 | . . . . . 6 |
53 | 52 | sselda 3603 | . . . . 5 |
54 | 53 | recnd 10068 | . . . 4 |
55 | 3, 6 | i1faddlem 23460 | . . . 4 |
56 | 54, 55 | syldan 487 | . . 3 |
57 | 16 | adantr 481 | . . . 4 |
58 | 3 | ad2antrr 762 | . . . . . . . 8 |
59 | i1fmbf 23442 | . . . . . . . 8 MblFn | |
60 | 58, 59 | syl 17 | . . . . . . 7 MblFn |
61 | 5 | ad2antrr 762 | . . . . . . 7 |
62 | 12 | ad2antrr 762 | . . . . . . . . . 10 |
63 | 62, 50 | syl 17 | . . . . . . . . 9 |
64 | eldifi 3732 | . . . . . . . . . 10 | |
65 | 64 | ad2antlr 763 | . . . . . . . . 9 |
66 | 63, 65 | sseldd 3604 | . . . . . . . 8 |
67 | 8 | adantr 481 | . . . . . . . . . 10 |
68 | frn 6053 | . . . . . . . . . 10 | |
69 | 67, 68 | syl 17 | . . . . . . . . 9 |
70 | 69 | sselda 3603 | . . . . . . . 8 |
71 | 66, 70 | resubcld 10458 | . . . . . . 7 |
72 | mbfimasn 23401 | . . . . . . 7 MblFn | |
73 | 60, 61, 71, 72 | syl3anc 1326 | . . . . . 6 |
74 | 6 | ad2antrr 762 | . . . . . . . 8 |
75 | i1fmbf 23442 | . . . . . . . 8 MblFn | |
76 | 74, 75 | syl 17 | . . . . . . 7 MblFn |
77 | 8 | ad2antrr 762 | . . . . . . 7 |
78 | mbfimasn 23401 | . . . . . . 7 MblFn | |
79 | 76, 77, 70, 78 | syl3anc 1326 | . . . . . 6 |
80 | inmbl 23310 | . . . . . 6 | |
81 | 73, 79, 80 | syl2anc 693 | . . . . 5 |
82 | 81 | ralrimiva 2966 | . . . 4 |
83 | finiunmbl 23312 | . . . 4 | |
84 | 57, 82, 83 | syl2anc 693 | . . 3 |
85 | 56, 84 | eqeltrd 2701 | . 2 |
86 | mblvol 23298 | . . . 4 | |
87 | 85, 86 | syl 17 | . . 3 |
88 | mblss 23299 | . . . . 5 | |
89 | 85, 88 | syl 17 | . . . 4 |
90 | inss1 3833 | . . . . . . . . 9 | |
91 | 90 | a1i 11 | . . . . . . . 8 |
92 | 73 | adantrr 753 | . . . . . . . . 9 |
93 | mblss 23299 | . . . . . . . . 9 | |
94 | 92, 93 | syl 17 | . . . . . . . 8 |
95 | mblvol 23298 | . . . . . . . . . 10 | |
96 | 92, 95 | syl 17 | . . . . . . . . 9 |
97 | simprr 796 | . . . . . . . . . . . . . . 15 | |
98 | 97 | oveq2d 6666 | . . . . . . . . . . . . . 14 |
99 | 54 | adantr 481 | . . . . . . . . . . . . . . 15 |
100 | 99 | subid1d 10381 | . . . . . . . . . . . . . 14 |
101 | 98, 100 | eqtrd 2656 | . . . . . . . . . . . . 13 |
102 | 101 | sneqd 4189 | . . . . . . . . . . . 12 |
103 | 102 | imaeq2d 5466 | . . . . . . . . . . 11 |
104 | 103 | fveq2d 6195 | . . . . . . . . . 10 |
105 | i1fima2sn 23447 | . . . . . . . . . . . 12 | |
106 | 3, 105 | sylan 488 | . . . . . . . . . . 11 |
107 | 106 | adantr 481 | . . . . . . . . . 10 |
108 | 104, 107 | eqeltrd 2701 | . . . . . . . . 9 |
109 | 96, 108 | eqeltrrd 2702 | . . . . . . . 8 |
110 | ovolsscl 23254 | . . . . . . . 8 | |
111 | 91, 94, 109, 110 | syl3anc 1326 | . . . . . . 7 |
112 | 111 | expr 643 | . . . . . 6 |
113 | eldifsn 4317 | . . . . . . . 8 | |
114 | inss2 3834 | . . . . . . . . . 10 | |
115 | 114 | a1i 11 | . . . . . . . . 9 |
116 | eldifi 3732 | . . . . . . . . . 10 | |
117 | mblss 23299 | . . . . . . . . . . 11 | |
118 | 79, 117 | syl 17 | . . . . . . . . . 10 |
119 | 116, 118 | sylan2 491 | . . . . . . . . 9 |
120 | i1fima 23445 | . . . . . . . . . . . . 13 | |
121 | 6, 120 | syl 17 | . . . . . . . . . . . 12 |
122 | 121 | ad2antrr 762 | . . . . . . . . . . 11 |
123 | mblvol 23298 | . . . . . . . . . . 11 | |
124 | 122, 123 | syl 17 | . . . . . . . . . 10 |
125 | 6 | adantr 481 | . . . . . . . . . . 11 |
126 | i1fima2sn 23447 | . . . . . . . . . . 11 | |
127 | 125, 126 | sylan 488 | . . . . . . . . . 10 |
128 | 124, 127 | eqeltrrd 2702 | . . . . . . . . 9 |
129 | ovolsscl 23254 | . . . . . . . . 9 | |
130 | 115, 119, 128, 129 | syl3anc 1326 | . . . . . . . 8 |
131 | 113, 130 | sylan2br 493 | . . . . . . 7 |
132 | 131 | expr 643 | . . . . . 6 |
133 | 112, 132 | pm2.61dne 2880 | . . . . 5 |
134 | 57, 133 | fsumrecl 14465 | . . . 4 |
135 | 56 | fveq2d 6195 | . . . . 5 |
136 | 114, 118 | syl5ss 3614 | . . . . . . . 8 |
137 | 136, 133 | jca 554 | . . . . . . 7 |
138 | 137 | ralrimiva 2966 | . . . . . 6 |
139 | ovolfiniun 23269 | . . . . . 6 | |
140 | 57, 138, 139 | syl2anc 693 | . . . . 5 |
141 | 135, 140 | eqbrtrd 4675 | . . . 4 |
142 | ovollecl 23251 | . . . 4 | |
143 | 89, 134, 141, 142 | syl3anc 1326 | . . 3 |
144 | 87, 143 | eqeltrd 2701 | . 2 |
145 | 12, 49, 85, 144 | i1fd 23448 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cab 2608 wne 2794 wral 2912 wrex 2913 cvv 3200 cdif 3571 cin 3573 wss 3574 csn 4177 ciun 4520 class class class wbr 4653 cxp 5112 ccnv 5113 cdm 5114 crn 5115 cima 5117 wfn 5883 wf 5884 wfo 5886 cfv 5888 (class class class)co 6650 cmpt2 6652 cof 6895 cfn 7955 cc 9934 cr 9935 cc0 9936 caddc 9939 cle 10075 cmin 10266 csu 14416 covol 23231 cvol 23232 MblFncmbf 23383 citg1 23384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-xmet 19739 df-met 19740 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 |
This theorem is referenced by: itg1addlem4 23466 i1fsub 23475 itg2splitlem 23515 itg2split 23516 itg2addlem 23525 itg2addnc 33464 ftc1anclem3 33487 ftc1anclem5 33489 ftc1anclem8 33492 |
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