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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8e | Structured version Visualization version GIF version | ||
| Description: Part of Part (8) in [Baer] p. 48. Eliminate 𝑤. (Contributed by NM, 10-May-2015.) | 
| Ref | Expression | 
|---|---|
| mapdh8a.h | ⊢ 𝐻 = (LHyp‘𝐾) | 
| mapdh8a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | 
| mapdh8a.v | ⊢ 𝑉 = (Base‘𝑈) | 
| mapdh8a.s | ⊢ − = (-g‘𝑈) | 
| mapdh8a.o | ⊢ 0 = (0g‘𝑈) | 
| mapdh8a.n | ⊢ 𝑁 = (LSpan‘𝑈) | 
| mapdh8a.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | 
| mapdh8a.d | ⊢ 𝐷 = (Base‘𝐶) | 
| mapdh8a.r | ⊢ 𝑅 = (-g‘𝐶) | 
| mapdh8a.q | ⊢ 𝑄 = (0g‘𝐶) | 
| mapdh8a.j | ⊢ 𝐽 = (LSpan‘𝐶) | 
| mapdh8a.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | 
| mapdh8a.i | ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) | 
| mapdh8a.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| mapdh8e.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) | 
| mapdh8e.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | 
| mapdh8e.eg | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | 
| mapdh8e.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | 
| mapdh8e.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | 
| mapdh8e.t | ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) | 
| mapdh8e.xy | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | 
| mapdh8e.xt | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) | 
| mapdh8e.yt | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) | 
| mapdh8e.e | ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) | 
| Ref | Expression | 
|---|---|
| mapdh8e | ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mapdh8a.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 2 | mapdh8a.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 3 | mapdh8a.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
| 4 | mapdh8a.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 5 | mapdh8a.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 6 | mapdh8e.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
| 7 | 6 | eldifad 3962 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | 
| 8 | mapdh8e.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
| 9 | 8 | eldifad 3962 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | 
| 10 | 1, 2, 3, 4, 5, 7, 9 | dvh3dim 41449 | . 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | 
| 11 | mapdh8a.s | . . . 4 ⊢ − = (-g‘𝑈) | |
| 12 | mapdh8a.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
| 13 | mapdh8a.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
| 14 | mapdh8a.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
| 15 | mapdh8a.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
| 16 | mapdh8a.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
| 17 | mapdh8a.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
| 18 | mapdh8a.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
| 19 | mapdh8a.i | . . . 4 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) | |
| 20 | 5 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 21 | mapdh8e.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
| 22 | 21 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐹 ∈ 𝐷) | 
| 23 | mapdh8e.mn | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
| 24 | 23 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | 
| 25 | mapdh8e.eg | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
| 26 | 25 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | 
| 27 | 6 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ (𝑉 ∖ { 0 })) | 
| 28 | 8 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ (𝑉 ∖ { 0 })) | 
| 29 | mapdh8e.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) | |
| 30 | 29 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑇 ∈ (𝑉 ∖ { 0 })) | 
| 31 | mapdh8e.yt | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) | |
| 32 | 31 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) | 
| 33 | eqid 2736 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
| 34 | 1, 2, 5 | dvhlmod 41113 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) | 
| 35 | 34 | 3ad2ant1 1133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod) | 
| 36 | 3, 33, 4, 34, 7, 9 | lspprcl 20977 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) | 
| 37 | 36 | 3ad2ant1 1133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) | 
| 38 | simp2 1137 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ 𝑉) | |
| 39 | simp3 1138 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
| 40 | 12, 33, 35, 37, 38, 39 | lssneln0 20952 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ (𝑉 ∖ { 0 })) | 
| 41 | 1, 2, 5 | dvhlvec 41112 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LVec) | 
| 42 | 29 | eldifad 3962 | . . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | 
| 43 | mapdh8e.xy | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
| 44 | mapdh8e.e | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) | |
| 45 | prcom 4731 | . . . . . . . . . . 11 ⊢ {𝑌, 𝑇} = {𝑇, 𝑌} | |
| 46 | 45 | fveq2i 6908 | . . . . . . . . . 10 ⊢ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑇, 𝑌}) | 
| 47 | 44, 46 | eleqtrdi 2850 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑇, 𝑌})) | 
| 48 | 3, 12, 4, 41, 6, 42, 9, 43, 47 | lspexch 21132 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝑋, 𝑌})) | 
| 49 | 33, 4, 34, 36, 48 | ellspsn5 20995 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑌})) | 
| 50 | 49 | 3ad2ant1 1133 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑌})) | 
| 51 | 34 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑈 ∈ LMod) | 
| 52 | 36 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) | 
| 53 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑤 ∈ 𝑉) | |
| 54 | 3, 33, 4, 51, 52, 53 | ellspsn5b 20994 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → (𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}))) | 
| 55 | 54 | biimprd 248 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → ((𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}) → 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))) | 
| 56 | 55 | con3d 152 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → ¬ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}))) | 
| 57 | 56 | 3impia 1117 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌})) | 
| 58 | nssne2 4046 | . . . . . 6 ⊢ (((𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑌}) ∧ ¬ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑇}) ≠ (𝑁‘{𝑤})) | |
| 59 | 50, 57, 58 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑇}) ≠ (𝑁‘{𝑤})) | 
| 60 | 59 | necomd 2995 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) | 
| 61 | mapdh8e.xt | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) | |
| 62 | 61 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) | 
| 63 | 41 | 3ad2ant1 1133 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LVec) | 
| 64 | 7 | 3ad2ant1 1133 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ 𝑉) | 
| 65 | 9 | 3ad2ant1 1133 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ 𝑉) | 
| 66 | 3, 4, 63, 38, 64, 65, 39 | lspindpi 21135 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) | 
| 67 | 66 | simprd 495 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) | 
| 68 | 67 | necomd 2995 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) | 
| 69 | 43 | 3ad2ant1 1133 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | 
| 70 | 3, 12, 4, 63, 27, 65, 38, 69, 39 | lspindp2l 21137 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ((𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))) | 
| 71 | 70 | simprd 495 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) | 
| 72 | 1, 2, 3, 11, 12, 4, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 27, 28, 30, 32, 40, 60, 62, 68, 71 | mapdh8d 41786 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) | 
| 73 | 72 | rexlimdv3a 3158 | . 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉))) | 
| 74 | 10, 73 | mpd 15 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑋, 𝐹, 𝑇〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∃wrex 3069 Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 ifcif 4524 {csn 4625 {cpr 4627 〈cotp 4633 ↦ cmpt 5224 ‘cfv 6560 ℩crio 7388 (class class class)co 7432 1st c1st 8013 2nd c2nd 8014 Basecbs 17248 0gc0g 17485 -gcsg 18954 LModclmod 20859 LSubSpclss 20930 LSpanclspn 20970 LVecclvec 21102 HLchlt 39352 LHypclh 39987 DVecHcdvh 41081 LCDualclcd 41589 mapdcmpd 41627 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-riotaBAD 38955 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-undef 8299 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-cntz 19336 df-oppg 19365 df-lsm 19655 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-nzr 20514 df-rlreg 20695 df-domn 20696 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lvec 21103 df-lsatoms 38978 df-lshyp 38979 df-lcv 39021 df-lfl 39060 df-lkr 39088 df-ldual 39126 df-oposet 39178 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-atl 39300 df-cvlat 39324 df-hlat 39353 df-llines 39501 df-lplanes 39502 df-lvols 39503 df-lines 39504 df-psubsp 39506 df-pmap 39507 df-padd 39799 df-lhyp 39991 df-laut 39992 df-ldil 40107 df-ltrn 40108 df-trl 40162 df-tgrp 40746 df-tendo 40758 df-edring 40760 df-dveca 41006 df-disoa 41032 df-dvech 41082 df-dib 41142 df-dic 41176 df-dih 41232 df-doch 41351 df-djh 41398 df-lcdual 41590 df-mapd 41628 | 
| This theorem is referenced by: mapdh8g 41788 | 
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