![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 7120 | . . . . 5 ⊢ (𝑌 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌) |
4 | 3 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌) |
5 | pmtridf1o.t | . . . . 5 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) | |
6 | simpr 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
7 | 6 | iftrued 4495 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴)) |
8 | 5, 7 | eqtrid 2785 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴)) |
9 | 8 | fveq1d 6845 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = (( I ↾ 𝐴)‘𝑌)) |
10 | 4, 9, 6 | 3eqtr4d 2783 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = 𝑋) |
11 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
12 | 11 | neneqd 2945 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌) |
13 | 12 | iffalsed 4498 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
14 | 5, 13 | eqtrid 2785 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
15 | 14 | fveq1d 6845 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌)) |
16 | pmtridf1o.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
17 | 16 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉) |
18 | pmtridf1o.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | 18 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴) |
20 | 1 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴) |
21 | eqid 2733 | . . . . 5 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴) | |
22 | 21 | pmtrprfv2 31988 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
23 | 17, 19, 20, 11, 22 | syl13anc 1373 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
24 | 15, 23 | eqtrd 2773 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = 𝑋) |
25 | 10, 24 | pm2.61dane 3029 | 1 ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ifcif 4487 {cpr 4589 I cid 5531 ↾ cres 5636 ‘cfv 6497 pmTrspcpmtr 19228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-1o 8413 df-2o 8414 df-en 8887 df-pmtr 19229 |
This theorem is referenced by: reprpmtf1o 33296 |
Copyright terms: Public domain | W3C validator |