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Theorem ivthinclemloc 15665
Description: Lemma for ivthinc 15667. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
Hypotheses
Ref Expression
ivth.1 |- ( ph -> A e. RR )
ivth.2 |- ( ph -> B e. RR )
ivth.3 |- ( ph -> U e. RR )
ivth.4 |- ( ph -> A < B )
ivth.5 |- ( ph -> ( A [,] B ) C_ D )
ivth.7 |- ( ph -> F e. ( D -cn-> CC ) )
ivth.8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
ivth.9 |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
ivthinc.i |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
ivthinclem.l |- L = { w e. ( A [,] B ) | ( F ` w ) < U }
ivthinclem.r |- R = { w e. ( A [,] B ) | U < ( F ` w ) }
Assertion
Ref Expression
ivthinclemloc |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. R ) ) )
Distinct variable groups:   A, r, w   x, A, y, r   B, r, w   x, B, y   w, F   x, F, y   w, U   ph, q, r, x, y   w, q
Allowed substitution hints:   ph( w)   A( q)   B( q)   D( x, y, w, r, q)   R( x, y, w, r, q)   U( x, y, r, q)   F( r, q)   L( x, y, w, r, q)
Proof of Theorem ivthinclemloc
StepHypRef Expression
1 simpr 110 . . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> q < r )
2 breq2 4129 . . . . . . . 8 |- ( y = r -> ( q < y <-> q < r ) )
3 fveq2 5690 . . . . . . . . 9 |- ( y = r -> ( F ` y ) = ( F ` r ) )
43breq2d 4137 . . . . . . . 8 |- ( y = r -> ( ( F ` q ) < ( F ` y ) <-> ( F ` q ) < ( F ` r ) ) )
52, 4imbi12d 234 . . . . . . 7 |- ( y = r -> ( ( q < y -> ( F ` q ) < ( F ` y ) ) <-> ( q < r -> ( F ` q ) < ( F ` r ) ) ) )
6 breq1 4128 . . . . . . . . . 10 |- ( x = q -> ( x < y <-> q < y ) )
7 fveq2 5690 . . . . . . . . . . 11 |- ( x = q -> ( F ` x ) = ( F ` q ) )
87breq1d 4135 . . . . . . . . . 10 |- ( x = q -> ( ( F ` x ) < ( F ` y ) <-> ( F ` q ) < ( F ` y ) ) )
96, 8imbi12d 234 . . . . . . . . 9 |- ( x = q -> ( ( x < y -> ( F ` x ) < ( F ` y ) ) <-> ( q < y -> ( F ` q ) < ( F ` y ) ) ) )
109ralbidv 2550 . . . . . . . 8 |- ( x = q -> ( A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) <-> A. y e. ( A [,] B ) ( q < y -> ( F ` q ) < ( F ` y ) ) ) )
11 ivthinc.i . . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )
1211expr 375 . . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y e. ( A [,] B ) ) -> ( x < y -> ( F ` x ) < ( F ` y ) ) )
1312ralrimiva 2623 . . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
1413ralrimiva 2623 . . . . . . . . 9 |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
1514ad2antrr 492 . . . . . . . 8 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( F ` x ) < ( F ` y ) ) )
16 simplrl 541 . . . . . . . 8 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> q e. ( A [,] B ) )
1710, 15, 16rspcdva 2934 . . . . . . 7 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> A. y e. ( A [,] B ) ( q < y -> ( F ` q ) < ( F ` y ) ) )
18 simplrr 542 . . . . . . 7 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> r e. ( A [,] B ) )
195, 17, 18rspcdva 2934 . . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( q < r -> ( F ` q ) < ( F ` r ) ) )
201, 19mpd 13 . . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( F ` q ) < ( F ` r ) )
217eleq1d 2307 . . . . . . 7 |- ( x = q -> ( ( F ` x ) e. RR <-> ( F ` q ) e. RR ) )
22 ivth.8 . . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
2322ralrimiva 2623 . . . . . . . 8 |- ( ph -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
2423ad2antrr 492 . . . . . . 7 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> A. x e. ( A [,] B ) ( F ` x ) e. RR )
2521, 24, 16rspcdva 2934 . . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( F ` q ) e. RR )
26 fveq2 5690 . . . . . . . 8 |- ( x = r -> ( F ` x ) = ( F ` r ) )
2726eleq1d 2307 . . . . . . 7 |- ( x = r -> ( ( F ` x ) e. RR <-> ( F ` r ) e. RR ) )
2827, 24, 18rspcdva 2934 . . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( F ` r ) e. RR )
29 ivth.3 . . . . . . 7 |- ( ph -> U e. RR )
3029ad2antrr 492 . . . . . 6 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> U e. RR )
31 axltwlin 8383 . . . . . 6 |- ( ( ( F ` q ) e. RR /\ ( F ` r ) e. RR /\ U e. RR ) -> ( ( F ` q ) < ( F ` r ) -> ( ( F ` q ) < U \/ U < ( F ` r ) ) ) )
3225, 28, 30, 31syl3anc 1278 . . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( ( F ` q ) < ( F ` r ) -> ( ( F ` q ) < U \/ U < ( F ` r ) ) ) )
3320, 32mpd 13 . . . 4 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( ( F ` q ) < U \/ U < ( F ` r ) ) )
3416adantr 276 . . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ ( F ` q ) < U ) -> q e. ( A [,] B ) )
35 simpr 110 . . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ ( F ` q ) < U ) -> ( F ` q ) < U )
36 fveq2 5690 . . . . . . . . 9 |- ( w = q -> ( F ` w ) = ( F ` q ) )
3736breq1d 4135 . . . . . . . 8 |- ( w = q -> ( ( F ` w ) < U <-> ( F ` q ) < U ) )
38 ivthinclem.l . . . . . . . 8 |- L = { w e. ( A [,] B ) | ( F ` w ) < U }
3937, 38elrab2 2985 . . . . . . 7 |- ( q e. L <-> ( q e. ( A [,] B ) /\ ( F ` q ) < U ) )
4034, 35, 39sylanbrc 421 . . . . . 6 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ ( F ` q ) < U ) -> q e. L )
4140ex 115 . . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( ( F ` q ) < U -> q e. L ) )
4218adantr 276 . . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ U < ( F ` r ) ) -> r e. ( A [,] B ) )
43 simpr 110 . . . . . . 7 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ U < ( F ` r ) ) -> U < ( F ` r ) )
44 fveq2 5690 . . . . . . . . 9 |- ( w = r -> ( F ` w ) = ( F ` r ) )
4544breq2d 4137 . . . . . . . 8 |- ( w = r -> ( U < ( F ` w ) <-> U < ( F ` r ) ) )
46 ivthinclem.r . . . . . . . 8 |- R = { w e. ( A [,] B ) | U < ( F ` w ) }
4745, 46elrab2 2985 . . . . . . 7 |- ( r e. R <-> ( r e. ( A [,] B ) /\ U < ( F ` r ) ) )
4842, 43, 47sylanbrc 421 . . . . . 6 |- ( ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) /\ U < ( F ` r ) ) -> r e. R )
4948ex 115 . . . . 5 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( U < ( F ` r ) -> r e. R ) )
5041, 49orim12d 798 . . . 4 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( ( ( F ` q ) < U \/ U < ( F ` r ) ) -> ( q e. L \/ r e. R ) ) )
5133, 50mpd 13 . . 3 |- ( ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) /\ q < r ) -> ( q e. L \/ r e. R ) )
5251ex 115 . 2 |- ( ( ph /\ ( q e. ( A [,] B ) /\ r e. ( A [,] B ) ) ) -> ( q < r -> ( q e. L \/ r e. R ) ) )
5352ralrimivva 2632 1 |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. R ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   C_ wss 3220   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   < clt 8350   [,]cicc 10272   -cn->ccncf 15594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltwlin 8282
This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-iota 5332  df-fv 5380  df-pnf 8352  df-mnf 8353  df-ltxr 8355
This theorem is referenced by:  ivthinclemex  15666
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