| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6aN | Structured version Visualization version GIF version | ||
| Description: Lemma for mapdh6N 42579. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| mapdh.q | ⊢ 𝑄 = (0g‘𝐶) |
| mapdh.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)}))))) |
| mapdh.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| mapdh.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
| mapdh.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| mapdh.v | ⊢ 𝑉 = (Base‘𝑈) |
| mapdh.s | ⊢ − = (-g‘𝑈) |
| mapdhc.o | ⊢ 0 = (0g‘𝑈) |
| mapdh.n | ⊢ 𝑁 = (LSpan‘𝑈) |
| mapdh.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
| mapdh.d | ⊢ 𝐷 = (Base‘𝐶) |
| mapdh.r | ⊢ 𝑅 = (-g‘𝐶) |
| mapdh.j | ⊢ 𝐽 = (LSpan‘𝐶) |
| mapdh.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| mapdhc.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
| mapdh.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
| mapdhcl.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
| mapdh.p | ⊢ + = (+g‘𝑈) |
| mapdh.a | ⊢ ✚ = (+g‘𝐶) |
| mapdhe6.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
| mapdhe6.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
| mapdhe6.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
| mapdh6.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
| mapdh6.fg | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
| mapdh6.fe | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) |
| Ref | Expression |
|---|---|
| mapdh6aN | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapdh.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
| 2 | mapdh.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)}))))) | |
| 3 | mapdh.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 4 | mapdh.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
| 5 | mapdh.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 6 | mapdh.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
| 7 | mapdh.s | . . . 4 ⊢ − = (-g‘𝑈) | |
| 8 | mapdhc.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
| 9 | mapdh.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 10 | mapdh.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
| 11 | mapdh.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
| 12 | mapdh.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
| 13 | mapdh.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
| 14 | mapdh.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 15 | mapdhc.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
| 16 | mapdh.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
| 17 | mapdhcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
| 18 | mapdh.p | . . . 4 ⊢ + = (+g‘𝑈) | |
| 19 | mapdh.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
| 20 | mapdhe6.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
| 21 | mapdhe6.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
| 22 | mapdhe6.xn | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
| 23 | mapdh6.yz | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) | |
| 24 | mapdh6.fg | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
| 25 | mapdh6.fe | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) | |
| 26 | 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 | mapdh6lem2N 42566 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐽‘{(𝐺 ✚ 𝐸)})) |
| 27 | 24, 25 | oveq12d 7437 | . . . . 5 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐺 ✚ 𝐸)) |
| 28 | 27 | sneqd 4603 | . . . 4 ⊢ (𝜑 → {((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))} = {(𝐺 ✚ 𝐸)}) |
| 29 | 28 | fveq2d 6889 | . . 3 ⊢ (𝜑 → (𝐽‘{((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))}) = (𝐽‘{(𝐺 ✚ 𝐸)})) |
| 30 | 26, 29 | eqtr4d 2803 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐽‘{((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))})) |
| 31 | 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 | mapdh6lem1N 42565 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐽‘{(𝐹𝑅(𝐺 ✚ 𝐸))})) |
| 32 | 27 | oveq2d 7435 | . . . . 5 ⊢ (𝜑 → (𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) = (𝐹𝑅(𝐺 ✚ 𝐸))) |
| 33 | 32 | sneqd 4603 | . . . 4 ⊢ (𝜑 → {(𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))} = {(𝐹𝑅(𝐺 ✚ 𝐸))}) |
| 34 | 33 | fveq2d 6889 | . . 3 ⊢ (𝜑 → (𝐽‘{(𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))}) = (𝐽‘{(𝐹𝑅(𝐺 ✚ 𝐸))})) |
| 35 | 31, 34 | eqtr4d 2803 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐽‘{(𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))})) |
| 36 | 3, 5, 14 | dvhlmod 41942 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 37 | 20 | eldifad 3918 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
| 38 | 21 | eldifad 3918 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
| 39 | 6, 18 | lmodvacl 21046 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
| 40 | 36, 37, 38, 39 | syl3anc 1398 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
| 41 | 6, 18, 8, 9, 36, 37, 38, 23 | lmodindp1 21185 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 ) |
| 42 | eldifsn 4755 | . . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 )) | |
| 43 | 40, 41, 42 | sylanbrc 595 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 })) |
| 44 | 3, 10, 14 | lcdlmod 42424 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
| 45 | 3, 5, 14 | dvhlvec 41941 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) |
| 46 | 17 | eldifad 3918 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| 47 | 6, 8, 9, 45, 37, 21, 46, 23, 22 | lspindp2 21309 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))) |
| 48 | 47 | simpld 500 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
| 49 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 37, 48 | mapdhcl 42559 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
| 50 | 6, 8, 9, 45, 20, 38, 46, 23, 22 | lspindp1 21307 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))) |
| 51 | 50 | simpld 500 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
| 52 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 38, 51 | mapdhcl 42559 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
| 53 | 11, 19 | lmodvacl 21046 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
| 54 | 44, 49, 52, 53 | syl3anc 1398 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
| 55 | eqid 2765 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
| 56 | 6, 55, 9, 36, 37, 38 | lspprcl 21149 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈)) |
| 57 | 6, 18, 9, 36, 37, 38 | lspprvacl 21170 | . . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑁‘{𝑌, 𝑍})) |
| 58 | 55, 9, 36, 56, 57 | ellspsn5 21167 | . . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍})) |
| 59 | 6, 55, 9, 36, 56, 46 | ellspsn5b 21166 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
| 60 | 22, 59 | mtbid 327 | . . . . 5 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})) |
| 61 | nssne2 4001 | . . . . 5 ⊢ (((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍}) ∧ ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})) → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋})) | |
| 62 | 58, 60, 61 | syl2anc 596 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋})) |
| 63 | 62 | necomd 3015 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
| 64 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 43, 54, 63 | mapdheq 42560 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ↔ ((𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐽‘{((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))}) ∧ (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐽‘{(𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))})))) |
| 65 | 30, 35, 64 | mpbir2and 726 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 Vcvv 3457 ∖ cdif 3903 ⊆ wss 3906 ifcif 4489 {csn 4591 {cpr 4593 〈cotp 4599 ↦ cmpt 5194 ‘cfv 6540 ℩crio 7375 (class class class)co 7419 1st c1st 7990 2nd c2nd 7991 Basecbs 17291 +gcplusg 17332 0gc0g 17514 -gcsg 19046 LModclmod 21031 LSubSpclss 21102 LSpanclspn 21142 HLchlt 40182 LHypclh 40816 DVecHcdvh 41910 LCDualclcd 42418 mapdcmpd 42456 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-riotaBAD 39785 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-n0 12520 df-z 12607 df-uz 12879 df-fz 13552 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-sca 17348 df-vsca 17349 df-0g 17516 df-mre 17660 df-mrc 17661 df-acs 17663 df-proset 18372 df-poset 18391 df-plt 18406 df-lub 18422 df-glb 18423 df-join 18424 df-meet 18425 df-p0 18501 df-p1 18502 df-lat 18510 df-clat 18577 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-grp 19047 df-minusg 19048 df-sbg 19049 df-subg 19233 df-cntz 19431 df-oppg 19460 df-lsm 19750 df-cmn 19896 df-abl 19897 df-mgp 20261 df-rng 20275 df-ur 20308 df-ring 20361 df-oppr 20465 df-dvdsr 20485 df-unit 20486 df-invr 20516 df-dvr 20529 df-nzr 20660 df-rlreg 20843 df-domn 20844 df-drng 20879 df-lmod 21033 df-lss 21103 df-lsp 21143 df-lvec 21274 df-lsatoms 39808 df-lshyp 39809 df-lcv 39851 df-lfl 39890 df-lkr 39918 df-ldual 39956 df-oposet 40008 df-ol 40010 df-oml 40011 df-covers 40098 df-ats 40099 df-atl 40130 df-cvlat 40154 df-hlat 40183 df-llines 40330 df-lplanes 40331 df-lvols 40332 df-lines 40333 df-psubsp 40335 df-pmap 40336 df-padd 40628 df-lhyp 40820 df-laut 40821 df-ldil 40936 df-ltrn 40937 df-trl 40991 df-tgrp 41575 df-tendo 41587 df-edring 41589 df-dveca 41835 df-disoa 41861 df-dvech 41911 df-dib 41971 df-dic 42005 df-dih 42061 df-doch 42180 df-djh 42227 df-lcdual 42419 df-mapd 42457 |
| This theorem is used by: mapdh6dN 42571 mapdh6eN 42572 mapdh6fN 42573 mapdh6jN 42577 |
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