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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdhval2 | Structured version Visualization version GIF version |
Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) |
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)}))))) |
mapdh2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
mapdh2.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
mapdh2.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
mapdhval2 | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.q | . . 3 ⊢ 𝑄 = (0g‘𝐶) | |
2 | mapdh.i | . . 3 ⊢ 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 | mapdh2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
4 | mapdh2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
5 | mapdh2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
6 | 1, 2, 3, 4, 5 | mapdhval 38875 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))) |
7 | eldifsni 4722 | . . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
8 | 7 | neneqd 3021 | . . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 ) |
9 | iffalse 4476 | . . 3 ⊢ (¬ 𝑌 = 0 → if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) | |
10 | 5, 8, 9 | 3syl 18 | . 2 ⊢ (𝜑 → if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) |
11 | 6, 10 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 ifcif 4467 {csn 4567 〈cotp 4575 ↦ cmpt 5146 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 0gc0g 16713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-riota 7114 df-ov 7159 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: mapdhcl 38878 mapdheq 38879 |
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