Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmtridfv2 | Structured version Visualization version GIF version |
Description: Value at Y of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.) |
Ref | Expression |
---|---|
pmtridf1o.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
pmtridf1o.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
pmtridf1o.y | ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
pmtridf1o.t | ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
pmtridfv2 | ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtridf1o.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
2 | fvresi 6937 | . . . . 5 ⊢ (𝑌 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌) |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌) |
5 | pmtridf1o.t | . . . . 5 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) | |
6 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
7 | 6 | iftrued 4477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴)) |
8 | 5, 7 | syl5eq 2870 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴)) |
9 | 8 | fveq1d 6674 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = (( I ↾ 𝐴)‘𝑌)) |
10 | 4, 9, 6 | 3eqtr4d 2868 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = 𝑋) |
11 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
12 | 11 | neneqd 3023 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌) |
13 | 12 | iffalsed 4480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
14 | 5, 13 | syl5eq 2870 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
15 | 14 | fveq1d 6674 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌)) |
16 | pmtridf1o.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
17 | 16 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉) |
18 | pmtridf1o.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | 18 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴) |
20 | 1 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴) |
21 | eqid 2823 | . . . . 5 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴) | |
22 | 21 | pmtrprfv2 30734 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
23 | 17, 19, 20, 11, 22 | syl13anc 1368 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
24 | 15, 23 | eqtrd 2858 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = 𝑋) |
25 | 10, 24 | pm2.61dane 3106 | 1 ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ifcif 4469 {cpr 4571 I cid 5461 ↾ cres 5559 ‘cfv 6357 pmTrspcpmtr 18571 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1o 8104 df-2o 8105 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pmtr 18572 |
This theorem is referenced by: reprpmtf1o 31899 |
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