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 6927 | . . . . 5 ⊢ (𝑌 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌) |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌) |
5 | pmtridf1o.t | . . . . 5 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) | |
6 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
7 | 6 | iftrued 4471 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴)) |
8 | 5, 7 | syl5eq 2865 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴)) |
9 | 8 | fveq1d 6665 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = (( I ↾ 𝐴)‘𝑌)) |
10 | 4, 9, 6 | 3eqtr4d 2863 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = 𝑋) |
11 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
12 | 11 | neneqd 3018 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌) |
13 | 12 | iffalsed 4474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
14 | 5, 13 | syl5eq 2865 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
15 | 14 | fveq1d 6665 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌)) |
16 | pmtridf1o.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉) |
18 | pmtridf1o.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | 18 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴) |
20 | 1 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴) |
21 | eqid 2818 | . . . . 5 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴) | |
22 | 21 | pmtrprfv2 30659 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
23 | 17, 19, 20, 11, 22 | syl13anc 1364 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
24 | 15, 23 | eqtrd 2853 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = 𝑋) |
25 | 10, 24 | pm2.61dane 3101 | 1 ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ifcif 4463 {cpr 4559 I cid 5452 ↾ cres 5550 ‘cfv 6348 pmTrspcpmtr 18498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-1o 8091 df-2o 8092 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pmtr 18499 |
This theorem is referenced by: reprpmtf1o 31796 |
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