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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilslem | Structured version Visualization version GIF version |
Description: Lemma for hlhilsbase2 36252. (Contributed by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhilslem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhilslem.e | ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) |
hlhilslem.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhilslem.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hlhilslem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhilslem.f | ⊢ 𝐹 = Slot 𝑁 |
hlhilslem.1 | ⊢ 𝑁 ∈ ℕ |
hlhilslem.2 | ⊢ 𝑁 < 4 |
hlhilslem.c | ⊢ 𝐶 = (𝐹‘𝐸) |
Ref | Expression |
---|---|
hlhilslem | ⊢ (𝜑 → 𝐶 = (𝐹‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilslem.c | . . 3 ⊢ 𝐶 = (𝐹‘𝐸) | |
2 | hlhilslem.f | . . . . 5 ⊢ 𝐹 = Slot 𝑁 | |
3 | hlhilslem.1 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
4 | 2, 3 | ndxid 15716 | . . . 4 ⊢ 𝐹 = Slot (𝐹‘ndx) |
5 | 3 | nnrei 10906 | . . . . . 6 ⊢ 𝑁 ∈ ℝ |
6 | hlhilslem.2 | . . . . . 6 ⊢ 𝑁 < 4 | |
7 | 5, 6 | ltneii 10029 | . . . . 5 ⊢ 𝑁 ≠ 4 |
8 | 2, 3 | ndxarg 15715 | . . . . . 6 ⊢ (𝐹‘ndx) = 𝑁 |
9 | starvndx 15827 | . . . . . 6 ⊢ (*𝑟‘ndx) = 4 | |
10 | 8, 9 | neeq12i 2848 | . . . . 5 ⊢ ((𝐹‘ndx) ≠ (*𝑟‘ndx) ↔ 𝑁 ≠ 4) |
11 | 7, 10 | mpbir 220 | . . . 4 ⊢ (𝐹‘ndx) ≠ (*𝑟‘ndx) |
12 | 4, 11 | setsnid 15743 | . . 3 ⊢ (𝐹‘𝐸) = (𝐹‘(𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)) |
13 | 1, 12 | eqtri 2632 | . 2 ⊢ 𝐶 = (𝐹‘(𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)) |
14 | hlhilslem.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
15 | hlhilslem.u | . . . . 5 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
16 | hlhilslem.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | hlhilslem.e | . . . . 5 ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) | |
18 | eqid 2610 | . . . . 5 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
19 | eqid 2610 | . . . . 5 ⊢ (𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) | |
20 | 14, 15, 16, 17, 18, 19 | hlhilsca 36245 | . . . 4 ⊢ (𝜑 → (𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (Scalar‘𝑈)) |
21 | hlhilslem.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
22 | 20, 21 | syl6eqr 2662 | . . 3 ⊢ (𝜑 → (𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = 𝑅) |
23 | 22 | fveq2d 6107 | . 2 ⊢ (𝜑 → (𝐹‘(𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)) = (𝐹‘𝑅)) |
24 | 13, 23 | syl5eq 2656 | 1 ⊢ (𝜑 → 𝐶 = (𝐹‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 < clt 9953 ℕcn 10897 4c4 10949 ndxcnx 15692 sSet csts 15693 Slot cslot 15694 *𝑟cstv 15770 Scalarcsca 15771 HLchlt 33655 LHypclh 34288 EDRingcedring 35059 HGMapchg 36193 HLHilchlh 36242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-hlhil 36243 |
This theorem is referenced by: hlhilsbase 36249 hlhilsplus 36250 hlhilsmul 36251 |
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