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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 2735 | . . . . . 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 1132 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
15 | simp3 1137 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ (𝑉 ∖ { 0 })) | |
16 | 15 | eldifad 3975 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ 𝑉) |
17 | hdmap14lem12.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
18 | 17 | 3ad2ant1 1132 | . . . . . 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 41850 | . . . . 5 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) |
20 | simp3 1137 | . . . . . . 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 2735 | . . . . . . . . 9 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
22 | hdmap14lem12.o | . . . . . . . . 9 ⊢ 0 = (0g‘𝑈) | |
23 | eqid 2735 | . . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
24 | eqid 2735 | . . . . . . . . 9 ⊢ (+g‘𝐶) = (+g‘𝐶) | |
25 | simp11 1202 | . . . . . . . . . 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 1204 | . . . . . . . . 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 1136 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑔 ∈ 𝐴) | |
34 | simp12 1203 | . . . . . . . . 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 41861 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐺 = 𝑔) |
36 | 35 | oveq1d 7446 | . . . . . . 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 2778 | . . . . . 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 3157 | . . . . 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 1120 | . . 3 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → (𝑦 ∈ (𝑉 ∖ { 0 }) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) |
41 | 40 | ralrimiv 3143 | . 2 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) |
42 | oveq2 7439 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐹 · 𝑦) = (𝐹 · 𝑋)) | |
43 | 42 | fveq2d 6911 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 𝑋))) |
44 | fveq2 6907 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑆‘𝑦) = (𝑆‘𝑋)) | |
45 | 44 | oveq2d 7447 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘𝑋))) |
46 | 43, 45 | eqeq12d 2751 | . . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))) |
47 | 46 | rspcv 3618 | . . . 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 406 | . 2 ⊢ ((𝜑 ∧ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) |
50 | 41, 49 | impbida 801 | 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∖ cdif 3960 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 HDMapchdma 41775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608 df-hvmap 41740 df-hdmap1 41776 df-hdmap 41777 |
This theorem is referenced by: hdmap14lem13 41863 |
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