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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem39 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 39222. Eliminate 𝐽. (Contributed by NM, 11-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem38.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfrlem38.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfrlem38.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfrlem38.p | ⊢ + = (+g‘𝑈) |
lcfrlem38.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfrlem38.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfrlem38.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcfrlem38.q | ⊢ 𝑄 = (LSubSp‘𝐷) |
lcfrlem38.c | ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcfrlem38.e | ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) |
lcfrlem38.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem38.g | ⊢ (𝜑 → 𝐺 ∈ 𝑄) |
lcfrlem38.gs | ⊢ (𝜑 → 𝐺 ⊆ 𝐶) |
lcfrlem38.xe | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
lcfrlem38.ye | ⊢ (𝜑 → 𝑌 ∈ 𝐸) |
lcfrlem38.z | ⊢ 0 = (0g‘𝑈) |
lcfrlem38.x | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
lcfrlem38.y | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
lcfrlem38.sp | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfrlem38.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
lcfrlem38.b | ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) |
lcfrlem38.i | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
lcfrlem38.n | ⊢ (𝜑 → 𝐼 ≠ 0 ) |
Ref | Expression |
---|---|
lcfrlem39 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem38.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcfrlem38.o | . 2 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lcfrlem38.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lcfrlem38.p | . 2 ⊢ + = (+g‘𝑈) | |
5 | lcfrlem38.f | . 2 ⊢ 𝐹 = (LFnl‘𝑈) | |
6 | lcfrlem38.l | . 2 ⊢ 𝐿 = (LKer‘𝑈) | |
7 | lcfrlem38.d | . 2 ⊢ 𝐷 = (LDual‘𝑈) | |
8 | lcfrlem38.q | . 2 ⊢ 𝑄 = (LSubSp‘𝐷) | |
9 | lcfrlem38.c | . 2 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
10 | lcfrlem38.e | . 2 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) | |
11 | lcfrlem38.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | lcfrlem38.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑄) | |
13 | lcfrlem38.gs | . 2 ⊢ (𝜑 → 𝐺 ⊆ 𝐶) | |
14 | lcfrlem38.xe | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
15 | lcfrlem38.ye | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐸) | |
16 | lcfrlem38.z | . 2 ⊢ 0 = (0g‘𝑈) | |
17 | lcfrlem38.x | . 2 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
18 | lcfrlem38.y | . 2 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
19 | lcfrlem38.sp | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
20 | lcfrlem38.ne | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
21 | lcfrlem38.b | . 2 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
22 | lcfrlem38.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
23 | lcfrlem38.n | . 2 ⊢ (𝜑 → 𝐼 ≠ 0 ) | |
24 | eqid 2738 | . 2 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
25 | eqid 2738 | . 2 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
26 | eqid 2738 | . 2 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
27 | eqid 2738 | . 2 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
28 | oveq1 7177 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑗( ·𝑠 ‘𝑈)𝑥) = (𝑘( ·𝑠 ‘𝑈)𝑥)) | |
29 | 28 | oveq2d 7186 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥)) = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥))) |
30 | 29 | eqeq2d 2749 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥)) ↔ 𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥)))) |
31 | 30 | rexbidv 3207 | . . . . 5 ⊢ (𝑗 = 𝑘 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥)))) |
32 | 31 | cbvriotavw 7137 | . . . 4 ⊢ (℩𝑗 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥))) = (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥))) |
33 | 32 | mpteq2i 5122 | . . 3 ⊢ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑗 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥)))) = (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥)))) |
34 | 33 | mpteq2i 5122 | . 2 ⊢ (𝑥 ∈ ((Base‘𝑈) ∖ { 0 }) ↦ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑗 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥))))) = (𝑥 ∈ ((Base‘𝑈) ∖ { 0 }) ↦ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥))))) |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 34 | lcfrlem38 39217 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∃wrex 3054 {crab 3057 ∖ cdif 3840 ∩ cin 3842 ⊆ wss 3843 {csn 4516 {cpr 4518 ∪ ciun 4881 ↦ cmpt 5110 ‘cfv 6339 ℩crio 7126 (class class class)co 7170 Basecbs 16586 +gcplusg 16668 Scalarcsca 16671 ·𝑠 cvsca 16672 0gc0g 16816 LSubSpclss 19822 LSpanclspn 19862 LFnlclfn 36694 LKerclk 36722 LDualcld 36760 HLchlt 36987 LHypclh 37621 DVecHcdvh 38715 ocHcoch 38984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-riotaBAD 36590 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-undef 7968 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-0g 16818 df-mre 16960 df-mrc 16961 df-acs 16963 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-p1 17766 df-lat 17772 df-clat 17834 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-cntz 18565 df-oppg 18592 df-lsm 18879 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-drng 19623 df-lmod 19755 df-lss 19823 df-lsp 19863 df-lvec 19994 df-lsatoms 36613 df-lshyp 36614 df-lcv 36656 df-lfl 36695 df-lkr 36723 df-ldual 36761 df-oposet 36813 df-ol 36815 df-oml 36816 df-covers 36903 df-ats 36904 df-atl 36935 df-cvlat 36959 df-hlat 36988 df-llines 37135 df-lplanes 37136 df-lvols 37137 df-lines 37138 df-psubsp 37140 df-pmap 37141 df-padd 37433 df-lhyp 37625 df-laut 37626 df-ldil 37741 df-ltrn 37742 df-trl 37796 df-tgrp 38380 df-tendo 38392 df-edring 38394 df-dveca 38640 df-disoa 38666 df-dvech 38716 df-dib 38776 df-dic 38810 df-dih 38866 df-doch 38985 df-djh 39032 |
This theorem is referenced by: lcfrlem40 39219 |
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