Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1450 Structured version   Unicode version

Theorem bnj1450 29493
 Description: Technical lemma for bnj60 29505. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1450.1
bnj1450.2
bnj1450.3
bnj1450.4
bnj1450.5
bnj1450.6
bnj1450.7
bnj1450.8
bnj1450.9
bnj1450.10
bnj1450.11
bnj1450.12
bnj1450.13
bnj1450.14
bnj1450.15
bnj1450.16
bnj1450.17
bnj1450.18
bnj1450.19
bnj1450.20
bnj1450.21
bnj1450.22
bnj1450.23
Assertion
Ref Expression
bnj1450
Distinct variable groups:   ,,,,,   ,   ,   ,,,   ,,,,,   ,,,,,   ,   ,   ,
Allowed substitution hints:   (,,,,)   (,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,,)   (,,,)   (,,,,)   (,,,)   (,,,,)   (,,,,)   (,)   (,,,,)   (,,,,)   (,,,)   (,,,)   (,,,,)   (,,,,)

Proof of Theorem bnj1450
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bnj1450.19 . . . . . . . . 9
21simprbi 452 . . . . . . . 8
3 bnj1450.17 . . . . . . . . . 10
4 bnj1450.15 . . . . . . . . . . 11
5 fndm 5547 . . . . . . . . . . 11
64, 5syl 16 . . . . . . . . . 10
73, 6bnj832 29200 . . . . . . . . 9
81, 7bnj832 29200 . . . . . . . 8
92, 8eleqtrrd 2515 . . . . . . 7
10 bnj1450.10 . . . . . . . 8
1110dmeqi 5074 . . . . . . 7
129, 11syl6eleq 2528 . . . . . 6
13 bnj1450.9 . . . . . . . 8
1413bnj1317 29267 . . . . . . 7
1514bnj1400 29281 . . . . . 6
1612, 15syl6eleq 2528 . . . . 5
1716bnj1405 29282 . . . 4
18 bnj1450.20 . . . 4
19 bnj1450.1 . . . . 5
20 bnj1450.2 . . . . 5
21 bnj1450.3 . . . . 5
22 bnj1450.4 . . . . 5
23 bnj1450.5 . . . . 5
24 bnj1450.6 . . . . 5
25 bnj1450.7 . . . . 5
26 bnj1450.8 . . . . 5
27 bnj1450.11 . . . . 5
28 bnj1450.12 . . . . 5
29 bnj1450.13 . . . . 5
30 bnj1450.14 . . . . 5
31 bnj1450.16 . . . . 5
32 bnj1450.18 . . . . 5
3319, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1bnj1449 29491 . . . 4
3417, 18, 33bnj1521 29296 . . 3
3513bnj1436 29285 . . . . . . . . . 10
3618, 35bnj836 29203 . . . . . . . . 9
3719, 20, 21, 22, 26bnj1373 29473 . . . . . . . . . 10
3837rexbii 2732 . . . . . . . . 9
3936, 38sylib 190 . . . . . . . 8
4039bnj1196 29240 . . . . . . 7
41 3anass 941 . . . . . . 7
4240, 41bnj1198 29241 . . . . . 6
43 bnj1450.21 . . . . . . 7
44 bnj252 29141 . . . . . . 7
4543, 44bitri 242 . . . . . 6
4619, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18bnj1444 29486 . . . . . 6
4742, 45, 46bnj1340 29269 . . . . 5
4821bnj1436 29285 . . . . . . . . . . 11
4943, 48bnj771 29207 . . . . . . . . . 10
5049bnj1196 29240 . . . . . . . . 9
51 3anass 941 . . . . . . . . 9
5250, 51bnj1198 29241 . . . . . . . 8
53 bnj1450.22 . . . . . . . . 9
54 bnj252 29141 . . . . . . . . 9
5553, 54bitri 242 . . . . . . . 8
56 bnj1450.23 . . . . . . . . 9
5719, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18, 43, 53, 56bnj1445 29487 . . . . . . . 8
5852, 55, 57bnj1340 29269 . . . . . . 7
5953bnj1254 29255 . . . . . . . . . 10
60 fveq2 5731 . . . . . . . . . . . 12
61 id 21 . . . . . . . . . . . . . . 15
62 bnj602 29360 . . . . . . . . . . . . . . . 16
6362reseq2d 5149 . . . . . . . . . . . . . . 15
6461, 63opeq12d 3994 . . . . . . . . . . . . . 14
6564, 20, 563eqtr4g 2495 . . . . . . . . . . . . 13
6665fveq2d 5735 . . . . . . . . . . . 12
6760, 66eqeq12d 2452 . . . . . . . . . . 11
6867cbvralv 2934 . . . . . . . . . 10
6959, 68sylib 190 . . . . . . . . 9
7018simp3bi 975 . . . . . . . . . . . 12
7143, 70bnj769 29205 . . . . . . . . . . 11
7253, 71bnj769 29205 . . . . . . . . . 10
73 fndm 5547 . . . . . . . . . . 11
7453, 73bnj771 29207 . . . . . . . . . 10
7572, 74eleqtrd 2514 . . . . . . . . 9
7669, 75bnj1294 29263 . . . . . . . 8
7731bnj930 29214 . . . . . . . . . . . . . 14
783, 77bnj832 29200 . . . . . . . . . . . . 13
791, 78bnj832 29200 . . . . . . . . . . . 12
8018, 79bnj835 29202 . . . . . . . . . . 11
8143, 80bnj769 29205 . . . . . . . . . 10
8253, 81bnj769 29205 . . . . . . . . 9
8318simp2bi 974 . . . . . . . . . . . 12
8443, 83bnj769 29205 . . . . . . . . . . 11
8553, 84bnj769 29205 . . . . . . . . . 10
86 elssuni 4045 . . . . . . . . . . 11
8786, 10syl6sseqr 3397 . . . . . . . . . 10
88 ssun3 3514 . . . . . . . . . . 11
8988, 28syl6sseqr 3397 . . . . . . . . . 10
9085, 87, 893syl 19 . . . . . . . . 9
9182, 90, 72bnj1502 29293 . . . . . . . 8
9219bnj1517 29295 . . . . . . . . . . . . . . . 16
9353, 92bnj770 29206 . . . . . . . . . . . . . . 15
9462sseq1d 3377 . . . . . . . . . . . . . . . 16
9594cbvralv 2934 . . . . . . . . . . . . . . 15
9693, 95sylib 190 . . . . . . . . . . . . . 14
9796, 75bnj1294 29263 . . . . . . . . . . . . 13
9897, 74sseqtr4d 3387 . . . . . . . . . . . 12
9982, 90, 98bnj1503 29294 . . . . . . . . . . 11
10099opeq2d 3993 . . . . . . . . . 10
101100, 29, 563eqtr4g 2495 . . . . . . . . 9
102101fveq2d 5735 . . . . . . . 8
10376, 91, 1023eqtr4d 2480 . . . . . . 7
10458, 103bnj593 29187 . . . . . 6
10519, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1446 29488 . . . . . 6
106104, 105bnj1397 29280 . . . . 5
10747, 106bnj593 29187 . . . 4
10819, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1447 29489 . . . 4
109107, 108bnj1397 29280 . . 3
11034, 109bnj593 29187 . 2
11119, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29bnj1448 29490 . 2
112110, 111bnj1397 29280 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360   w3a 937  wex 1551   wceq 1653   wcel 1726  cab 2424   wne 2601  wral 2707  wrex 2708  crab 2711  wsbc 3163   cun 3320   wss 3322  c0 3630  csn 3816  cop 3819  cuni 4017  ciun 4095   class class class wbr 4215   cdm 4881   cres 4883   wfun 5451   wfn 5452  cfv 5457   w-bnj17 29124   c-bnj14 29126   w-bnj15 29130   c-bnj18 29132 This theorem is referenced by:  bnj1423  29494 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-bnj17 29125  df-bnj14 29127  df-bnj18 29133
 Copyright terms: Public domain W3C validator