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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 7193 | . . . . 5 ⊢ (𝑌 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌) |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌) |
5 | pmtridf1o.t | . . . . 5 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) | |
6 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
7 | 6 | iftrued 4539 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴)) |
8 | 5, 7 | eqtrid 2787 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴)) |
9 | 8 | fveq1d 6909 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = (( I ↾ 𝐴)‘𝑌)) |
10 | 4, 9, 6 | 3eqtr4d 2785 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = 𝑋) |
11 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
12 | 11 | neneqd 2943 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌) |
13 | 12 | iffalsed 4542 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
14 | 5, 13 | eqtrid 2787 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
15 | 14 | fveq1d 6909 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌)) |
16 | pmtridf1o.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
17 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉) |
18 | pmtridf1o.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴) |
20 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴) |
21 | eqid 2735 | . . . . 5 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴) | |
22 | 21 | pmtrprfv2 33091 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
23 | 17, 19, 20, 11, 22 | syl13anc 1371 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋) |
24 | 15, 23 | eqtrd 2775 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = 𝑋) |
25 | 10, 24 | pm2.61dane 3027 | 1 ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ifcif 4531 {cpr 4633 I cid 5582 ↾ cres 5691 ‘cfv 6563 pmTrspcpmtr 19474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1o 8505 df-2o 8506 df-en 8985 df-pmtr 19475 |
This theorem is referenced by: reprpmtf1o 34620 |
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