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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem12 | Structured version Visualization version GIF version |
Description: Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.) |
Ref | Expression |
---|---|
hdmap14lem12.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem12.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem12.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem12.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem12.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem12.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem12.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem12.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem12.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem12.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem12.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
hdmap14lem12.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem12.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem12.o | ⊢ 0 = (0g‘𝑈) |
hdmap14lem12.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem12.g | ⊢ (𝜑 → 𝐺 ∈ 𝐴) |
Ref | Expression |
---|---|
hdmap14lem12 | ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem12.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap14lem12.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap14lem12.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap14lem12.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑈) | |
5 | hdmap14lem12.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | hdmap14lem12.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
7 | hdmap14lem12.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap14lem12.e | . . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
9 | eqid 2797 | . . . . . 6 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
10 | hdmap14lem12.p | . . . . . 6 ⊢ 𝑃 = (Scalar‘𝐶) | |
11 | hdmap14lem12.a | . . . . . 6 ⊢ 𝐴 = (Base‘𝑃) | |
12 | hdmap14lem12.s | . . . . . 6 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
13 | hdmap14lem12.k | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | 13 | 3ad2ant1 1126 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
15 | simp3 1131 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ (𝑉 ∖ { 0 })) | |
16 | 15 | eldifad 3877 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ 𝑉) |
17 | hdmap14lem12.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
18 | 17 | 3ad2ant1 1126 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝐹 ∈ 𝐵) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18 | hdmap14lem2a 38555 | . . . . 5 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) |
20 | simp3 1131 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) | |
21 | eqid 2797 | . . . . . . . . 9 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
22 | hdmap14lem12.o | . . . . . . . . 9 ⊢ 0 = (0g‘𝑈) | |
23 | eqid 2797 | . . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
24 | eqid 2797 | . . . . . . . . 9 ⊢ (+g‘𝐶) = (+g‘𝐶) | |
25 | simp11 1196 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝜑) | |
26 | 25, 13 | syl 17 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
27 | hdmap14lem12.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
28 | 25, 27 | syl 17 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
29 | simp13 1198 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑦 ∈ (𝑉 ∖ { 0 })) | |
30 | 25, 17 | syl 17 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐹 ∈ 𝐵) |
31 | hdmap14lem12.g | . . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝐴) | |
32 | 25, 31 | syl 17 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐺 ∈ 𝐴) |
33 | simp2 1130 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑔 ∈ 𝐴) | |
34 | simp12 1197 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) | |
35 | 1, 2, 3, 21, 4, 22, 23, 5, 6, 7, 24, 8, 10, 11, 12, 26, 28, 29, 30, 32, 33, 34, 20 | hdmap14lem11 38566 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐺 = 𝑔) |
36 | 35 | oveq1d 7038 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝐺 ∙ (𝑆‘𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) |
37 | 20, 36 | eqtr4d 2836 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) |
38 | 37 | rexlimdv3a 3251 | . . . . 5 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦)) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
39 | 19, 38 | mpd 15 | . . . 4 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) |
40 | 39 | 3expia 1114 | . . 3 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → (𝑦 ∈ (𝑉 ∖ { 0 }) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
41 | 40 | ralrimiv 3150 | . 2 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) |
42 | oveq2 7031 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐹 · 𝑦) = (𝐹 · 𝑋)) | |
43 | 42 | fveq2d 6549 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 𝑋))) |
44 | fveq2 6545 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑆‘𝑦) = (𝑆‘𝑋)) | |
45 | 44 | oveq2d 7039 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘𝑋))) |
46 | 43, 45 | eqeq12d 2812 | . . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))) |
47 | 46 | rspcv 3557 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))) |
48 | 27, 47 | syl 17 | . . 3 ⊢ (𝜑 → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))) |
49 | 48 | imp 407 | . 2 ⊢ ((𝜑 ∧ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) |
50 | 41, 49 | impbida 797 | 1 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ∀wral 3107 ∃wrex 3108 ∖ cdif 3862 {csn 4478 ‘cfv 6232 (class class class)co 7023 Basecbs 16316 +gcplusg 16398 Scalarcsca 16401 ·𝑠 cvsca 16402 0gc0g 16546 LSpanclspn 19437 HLchlt 36038 LHypclh 36672 DVecHcdvh 37766 LCDualclcd 38274 HDMapchdma 38480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-riotaBAD 35641 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-ot 4487 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-tpos 7750 df-undef 7797 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-sca 16414 df-vsca 16415 df-0g 16548 df-mre 16690 df-mrc 16691 df-acs 16693 df-proset 17371 df-poset 17389 df-plt 17401 df-lub 17417 df-glb 17418 df-join 17419 df-meet 17420 df-p0 17482 df-p1 17483 df-lat 17489 df-clat 17551 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-grp 17868 df-minusg 17869 df-sbg 17870 df-subg 18034 df-cntz 18192 df-oppg 18219 df-lsm 18495 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-dvr 19127 df-drng 19198 df-lmod 19330 df-lss 19398 df-lsp 19438 df-lvec 19569 df-lsatoms 35664 df-lshyp 35665 df-lcv 35707 df-lfl 35746 df-lkr 35774 df-ldual 35812 df-oposet 35864 df-ol 35866 df-oml 35867 df-covers 35954 df-ats 35955 df-atl 35986 df-cvlat 36010 df-hlat 36039 df-llines 36186 df-lplanes 36187 df-lvols 36188 df-lines 36189 df-psubsp 36191 df-pmap 36192 df-padd 36484 df-lhyp 36676 df-laut 36677 df-ldil 36792 df-ltrn 36793 df-trl 36847 df-tgrp 37431 df-tendo 37443 df-edring 37445 df-dveca 37691 df-disoa 37717 df-dvech 37767 df-dib 37827 df-dic 37861 df-dih 37917 df-doch 38036 df-djh 38083 df-lcdual 38275 df-mapd 38313 df-hvmap 38445 df-hdmap1 38481 df-hdmap 38482 |
This theorem is referenced by: hdmap14lem13 38568 |
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