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Theorem psgnunilem1 18550
Description: Lemma for psgnuni 18556. Given two consequtive transpositions in a representation of a permutation, either they are equal and therefore equivalent to the identity, or they are not and it is possible to commute them such that a chosen point in the left transposition is preserved in the right. By repeating this process, a point can be removed from a representation of the identity. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
psgnunilem1.t 𝑇 = ran (pmTrsp‘𝐷)
psgnunilem1.d (𝜑𝐷𝑉)
psgnunilem1.p (𝜑𝑃𝑇)
psgnunilem1.q (𝜑𝑄𝑇)
psgnunilem1.a (𝜑𝐴 ∈ dom (𝑃 ∖ I ))
Assertion
Ref Expression
psgnunilem1 (𝜑 → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
Distinct variable groups:   𝑠,𝑟,𝐴   𝑃,𝑟,𝑠   𝑄,𝑟,𝑠   𝑇,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑠,𝑟)   𝐷(𝑠,𝑟)   𝑉(𝑠,𝑟)

Proof of Theorem psgnunilem1
StepHypRef Expression
1 psgnunilem1.q . . . . . . . 8 (𝜑𝑄𝑇)
2 eqid 2818 . . . . . . . . 9 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
3 psgnunilem1.t . . . . . . . . 9 𝑇 = ran (pmTrsp‘𝐷)
42, 3pmtrfinv 18518 . . . . . . . 8 (𝑄𝑇 → (𝑄𝑄) = ( I ↾ 𝐷))
51, 4syl 17 . . . . . . 7 (𝜑 → (𝑄𝑄) = ( I ↾ 𝐷))
6 coeq1 5721 . . . . . . . 8 (𝑃 = 𝑄 → (𝑃𝑄) = (𝑄𝑄))
76eqeq1d 2820 . . . . . . 7 (𝑃 = 𝑄 → ((𝑃𝑄) = ( I ↾ 𝐷) ↔ (𝑄𝑄) = ( I ↾ 𝐷)))
85, 7syl5ibrcom 248 . . . . . 6 (𝜑 → (𝑃 = 𝑄 → (𝑃𝑄) = ( I ↾ 𝐷)))
98adantr 481 . . . . 5 ((𝜑𝐴 ∈ dom (𝑄 ∖ I )) → (𝑃 = 𝑄 → (𝑃𝑄) = ( I ↾ 𝐷)))
109imp 407 . . . 4 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃 = 𝑄) → (𝑃𝑄) = ( I ↾ 𝐷))
1110orcd 869 . . 3 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃 = 𝑄) → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
12 psgnunilem1.p . . . . . . . . . 10 (𝜑𝑃𝑇)
132, 3pmtrfcnv 18521 . . . . . . . . . 10 (𝑃𝑇𝑃 = 𝑃)
1412, 13syl 17 . . . . . . . . 9 (𝜑𝑃 = 𝑃)
1514eqcomd 2824 . . . . . . . 8 (𝜑𝑃 = 𝑃)
1615coeq2d 5726 . . . . . . 7 (𝜑 → ((𝑃𝑄) ∘ 𝑃) = ((𝑃𝑄) ∘ 𝑃))
172, 3pmtrff1o 18520 . . . . . . . . 9 (𝑃𝑇𝑃:𝐷1-1-onto𝐷)
1812, 17syl 17 . . . . . . . 8 (𝜑𝑃:𝐷1-1-onto𝐷)
192, 3pmtrfconj 18523 . . . . . . . 8 ((𝑄𝑇𝑃:𝐷1-1-onto𝐷) → ((𝑃𝑄) ∘ 𝑃) ∈ 𝑇)
201, 18, 19syl2anc 584 . . . . . . 7 (𝜑 → ((𝑃𝑄) ∘ 𝑃) ∈ 𝑇)
2116, 20eqeltrd 2910 . . . . . 6 (𝜑 → ((𝑃𝑄) ∘ 𝑃) ∈ 𝑇)
2221ad2antrr 722 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ((𝑃𝑄) ∘ 𝑃) ∈ 𝑇)
2312ad2antrr 722 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → 𝑃𝑇)
24 coass 6111 . . . . . . 7 (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃) = ((𝑃𝑄) ∘ (𝑃𝑃))
252, 3pmtrfinv 18518 . . . . . . . . . 10 (𝑃𝑇 → (𝑃𝑃) = ( I ↾ 𝐷))
2612, 25syl 17 . . . . . . . . 9 (𝜑 → (𝑃𝑃) = ( I ↾ 𝐷))
2726coeq2d 5726 . . . . . . . 8 (𝜑 → ((𝑃𝑄) ∘ (𝑃𝑃)) = ((𝑃𝑄) ∘ ( I ↾ 𝐷)))
28 f1of 6608 . . . . . . . . . . 11 (𝑃:𝐷1-1-onto𝐷𝑃:𝐷𝐷)
2918, 28syl 17 . . . . . . . . . 10 (𝜑𝑃:𝐷𝐷)
302, 3pmtrff1o 18520 . . . . . . . . . . . 12 (𝑄𝑇𝑄:𝐷1-1-onto𝐷)
311, 30syl 17 . . . . . . . . . . 11 (𝜑𝑄:𝐷1-1-onto𝐷)
32 f1of 6608 . . . . . . . . . . 11 (𝑄:𝐷1-1-onto𝐷𝑄:𝐷𝐷)
3331, 32syl 17 . . . . . . . . . 10 (𝜑𝑄:𝐷𝐷)
34 fco 6524 . . . . . . . . . 10 ((𝑃:𝐷𝐷𝑄:𝐷𝐷) → (𝑃𝑄):𝐷𝐷)
3529, 33, 34syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑃𝑄):𝐷𝐷)
36 fcoi1 6545 . . . . . . . . 9 ((𝑃𝑄):𝐷𝐷 → ((𝑃𝑄) ∘ ( I ↾ 𝐷)) = (𝑃𝑄))
3735, 36syl 17 . . . . . . . 8 (𝜑 → ((𝑃𝑄) ∘ ( I ↾ 𝐷)) = (𝑃𝑄))
3827, 37eqtrd 2853 . . . . . . 7 (𝜑 → ((𝑃𝑄) ∘ (𝑃𝑃)) = (𝑃𝑄))
3924, 38syl5req 2866 . . . . . 6 (𝜑 → (𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃))
4039ad2antrr 722 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → (𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃))
41 psgnunilem1.a . . . . . 6 (𝜑𝐴 ∈ dom (𝑃 ∖ I ))
4241ad2antrr 722 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → 𝐴 ∈ dom (𝑃 ∖ I ))
4318adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝑃:𝐷1-1-onto𝐷)
4431adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝑄:𝐷1-1-onto𝐷)
452, 3pmtrfb 18522 . . . . . . . . . . . . 13 (𝑃𝑇 ↔ (𝐷 ∈ V ∧ 𝑃:𝐷1-1-onto𝐷 ∧ dom (𝑃 ∖ I ) ≈ 2o))
4645simp3bi 1139 . . . . . . . . . . . 12 (𝑃𝑇 → dom (𝑃 ∖ I ) ≈ 2o)
4712, 46syl 17 . . . . . . . . . . 11 (𝜑 → dom (𝑃 ∖ I ) ≈ 2o)
4847adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) ≈ 2o)
49 2onn 8255 . . . . . . . . . . . . . . 15 2o ∈ ω
50 nnfi 8699 . . . . . . . . . . . . . . 15 (2o ∈ ω → 2o ∈ Fin)
5149, 50ax-mp 5 . . . . . . . . . . . . . 14 2o ∈ Fin
522, 3pmtrfb 18522 . . . . . . . . . . . . . . . . 17 (𝑄𝑇 ↔ (𝐷 ∈ V ∧ 𝑄:𝐷1-1-onto𝐷 ∧ dom (𝑄 ∖ I ) ≈ 2o))
5352simp3bi 1139 . . . . . . . . . . . . . . . 16 (𝑄𝑇 → dom (𝑄 ∖ I ) ≈ 2o)
541, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → dom (𝑄 ∖ I ) ≈ 2o)
55 enfi 8722 . . . . . . . . . . . . . . 15 (dom (𝑄 ∖ I ) ≈ 2o → (dom (𝑄 ∖ I ) ∈ Fin ↔ 2o ∈ Fin))
5654, 55syl 17 . . . . . . . . . . . . . 14 (𝜑 → (dom (𝑄 ∖ I ) ∈ Fin ↔ 2o ∈ Fin))
5751, 56mpbiri 259 . . . . . . . . . . . . 13 (𝜑 → dom (𝑄 ∖ I ) ∈ Fin)
5857adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑄 ∖ I ) ∈ Fin)
5941adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝐴 ∈ dom (𝑃 ∖ I ))
60 en2eleq 9422 . . . . . . . . . . . . . 14 ((𝐴 ∈ dom (𝑃 ∖ I ) ∧ dom (𝑃 ∖ I ) ≈ 2o) → dom (𝑃 ∖ I ) = {𝐴, (dom (𝑃 ∖ I ) ∖ {𝐴})})
6159, 48, 60syl2anc 584 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) = {𝐴, (dom (𝑃 ∖ I ) ∖ {𝐴})})
62 simprl 767 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝐴 ∈ dom (𝑄 ∖ I ))
63 f1ofn 6609 . . . . . . . . . . . . . . . . . 18 (𝑃:𝐷1-1-onto𝐷𝑃 Fn 𝐷)
6418, 63syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 Fn 𝐷)
6564adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝑃 Fn 𝐷)
66 fimass 6548 . . . . . . . . . . . . . . . . . 18 (𝑃:𝐷𝐷 → (𝑃 “ dom (𝑄 ∖ I )) ⊆ 𝐷)
6729, 66syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃 “ dom (𝑄 ∖ I )) ⊆ 𝐷)
6867adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (𝑃 “ dom (𝑄 ∖ I )) ⊆ 𝐷)
69 simprr 769 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))
70 fnfvima 6986 . . . . . . . . . . . . . . . 16 ((𝑃 Fn 𝐷 ∧ (𝑃 “ dom (𝑄 ∖ I )) ⊆ 𝐷𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I ))) → (𝑃𝐴) ∈ (𝑃 “ (𝑃 “ dom (𝑄 ∖ I ))))
7165, 68, 69, 70syl3anc 1363 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (𝑃𝐴) ∈ (𝑃 “ (𝑃 “ dom (𝑄 ∖ I ))))
72 difss 4105 . . . . . . . . . . . . . . . . . . . . 21 (𝑃 ∖ I ) ⊆ 𝑃
73 dmss 5764 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∖ I ) ⊆ 𝑃 → dom (𝑃 ∖ I ) ⊆ dom 𝑃)
7472, 73ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 dom (𝑃 ∖ I ) ⊆ dom 𝑃
75 f1odm 6612 . . . . . . . . . . . . . . . . . . . . 21 (𝑃:𝐷1-1-onto𝐷 → dom 𝑃 = 𝐷)
7618, 75syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝑃 = 𝐷)
7774, 76sseqtrid 4016 . . . . . . . . . . . . . . . . . . 19 (𝜑 → dom (𝑃 ∖ I ) ⊆ 𝐷)
7877, 41sseldd 3965 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴𝐷)
79 eqid 2818 . . . . . . . . . . . . . . . . . . 19 dom (𝑃 ∖ I ) = dom (𝑃 ∖ I )
802, 3, 79pmtrffv 18516 . . . . . . . . . . . . . . . . . 18 ((𝑃𝑇𝐴𝐷) → (𝑃𝐴) = if(𝐴 ∈ dom (𝑃 ∖ I ), (dom (𝑃 ∖ I ) ∖ {𝐴}), 𝐴))
8112, 78, 80syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃𝐴) = if(𝐴 ∈ dom (𝑃 ∖ I ), (dom (𝑃 ∖ I ) ∖ {𝐴}), 𝐴))
8241iftrued 4471 . . . . . . . . . . . . . . . . 17 (𝜑 → if(𝐴 ∈ dom (𝑃 ∖ I ), (dom (𝑃 ∖ I ) ∖ {𝐴}), 𝐴) = (dom (𝑃 ∖ I ) ∖ {𝐴}))
8381, 82eqtrd 2853 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃𝐴) = (dom (𝑃 ∖ I ) ∖ {𝐴}))
8483adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (𝑃𝐴) = (dom (𝑃 ∖ I ) ∖ {𝐴}))
85 imaco 6097 . . . . . . . . . . . . . . . . 17 ((𝑃𝑃) “ dom (𝑄 ∖ I )) = (𝑃 “ (𝑃 “ dom (𝑄 ∖ I )))
8626imaeq1d 5921 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑃𝑃) “ dom (𝑄 ∖ I )) = (( I ↾ 𝐷) “ dom (𝑄 ∖ I )))
87 difss 4105 . . . . . . . . . . . . . . . . . . . . . 22 (𝑄 ∖ I ) ⊆ 𝑄
88 dmss 5764 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑄 ∖ I ) ⊆ 𝑄 → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
8987, 88ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 dom (𝑄 ∖ I ) ⊆ dom 𝑄
90 f1odm 6612 . . . . . . . . . . . . . . . . . . . . 21 (𝑄:𝐷1-1-onto𝐷 → dom 𝑄 = 𝐷)
9189, 90sseqtrid 4016 . . . . . . . . . . . . . . . . . . . 20 (𝑄:𝐷1-1-onto𝐷 → dom (𝑄 ∖ I ) ⊆ 𝐷)
9231, 91syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → dom (𝑄 ∖ I ) ⊆ 𝐷)
93 resiima 5937 . . . . . . . . . . . . . . . . . . 19 (dom (𝑄 ∖ I ) ⊆ 𝐷 → (( I ↾ 𝐷) “ dom (𝑄 ∖ I )) = dom (𝑄 ∖ I ))
9492, 93syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (( I ↾ 𝐷) “ dom (𝑄 ∖ I )) = dom (𝑄 ∖ I ))
9586, 94eqtrd 2853 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃𝑃) “ dom (𝑄 ∖ I )) = dom (𝑄 ∖ I ))
9685, 95syl5eqr 2867 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃 “ (𝑃 “ dom (𝑄 ∖ I ))) = dom (𝑄 ∖ I ))
9796adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (𝑃 “ (𝑃 “ dom (𝑄 ∖ I ))) = dom (𝑄 ∖ I ))
9871, 84, 973eltr3d 2924 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (dom (𝑃 ∖ I ) ∖ {𝐴}) ∈ dom (𝑄 ∖ I ))
9962, 98prssd 4747 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → {𝐴, (dom (𝑃 ∖ I ) ∖ {𝐴})} ⊆ dom (𝑄 ∖ I ))
10061, 99eqsstrd 4002 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) ⊆ dom (𝑄 ∖ I ))
10154ensymd 8548 . . . . . . . . . . . . . 14 (𝜑 → 2o ≈ dom (𝑄 ∖ I ))
102 entr 8549 . . . . . . . . . . . . . 14 ((dom (𝑃 ∖ I ) ≈ 2o ∧ 2o ≈ dom (𝑄 ∖ I )) → dom (𝑃 ∖ I ) ≈ dom (𝑄 ∖ I ))
10347, 101, 102syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → dom (𝑃 ∖ I ) ≈ dom (𝑄 ∖ I ))
104103adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) ≈ dom (𝑄 ∖ I ))
105 fisseneq 8717 . . . . . . . . . . . 12 ((dom (𝑄 ∖ I ) ∈ Fin ∧ dom (𝑃 ∖ I ) ⊆ dom (𝑄 ∖ I ) ∧ dom (𝑃 ∖ I ) ≈ dom (𝑄 ∖ I )) → dom (𝑃 ∖ I ) = dom (𝑄 ∖ I ))
10658, 100, 104, 105syl3anc 1363 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) = dom (𝑄 ∖ I ))
107106eqcomd 2824 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑄 ∖ I ) = dom (𝑃 ∖ I ))
108 f1otrspeq 18504 . . . . . . . . . 10 (((𝑃:𝐷1-1-onto𝐷𝑄:𝐷1-1-onto𝐷) ∧ (dom (𝑃 ∖ I ) ≈ 2o ∧ dom (𝑄 ∖ I ) = dom (𝑃 ∖ I ))) → 𝑃 = 𝑄)
10943, 44, 48, 107, 108syl22anc 834 . . . . . . . . 9 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝑃 = 𝑄)
110109expr 457 . . . . . . . 8 ((𝜑𝐴 ∈ dom (𝑄 ∖ I )) → (𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )) → 𝑃 = 𝑄))
111110necon3ad 3026 . . . . . . 7 ((𝜑𝐴 ∈ dom (𝑄 ∖ I )) → (𝑃𝑄 → ¬ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I ))))
112111imp 407 . . . . . 6 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ¬ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))
11316difeq1d 4095 . . . . . . . . . 10 (𝜑 → (((𝑃𝑄) ∘ 𝑃) ∖ I ) = (((𝑃𝑄) ∘ 𝑃) ∖ I ))
114113dmeqd 5767 . . . . . . . . 9 (𝜑 → dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) = dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))
115 f1omvdconj 18503 . . . . . . . . . 10 ((𝑄:𝐷𝐷𝑃:𝐷1-1-onto𝐷) → dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) = (𝑃 “ dom (𝑄 ∖ I )))
11633, 18, 115syl2anc 584 . . . . . . . . 9 (𝜑 → dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) = (𝑃 “ dom (𝑄 ∖ I )))
117114, 116eqtrd 2853 . . . . . . . 8 (𝜑 → dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) = (𝑃 “ dom (𝑄 ∖ I )))
118117eleq2d 2895 . . . . . . 7 (𝜑 → (𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) ↔ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I ))))
119118ad2antrr 722 . . . . . 6 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → (𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) ↔ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I ))))
120112, 119mtbird 326 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))
121 coeq1 5721 . . . . . . . 8 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (𝑟𝑠) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠))
122121eqeq2d 2829 . . . . . . 7 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → ((𝑃𝑄) = (𝑟𝑠) ↔ (𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠)))
123 difeq1 4089 . . . . . . . . . 10 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (𝑟 ∖ I ) = (((𝑃𝑄) ∘ 𝑃) ∖ I ))
124123dmeqd 5767 . . . . . . . . 9 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → dom (𝑟 ∖ I ) = dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))
125124eleq2d 2895 . . . . . . . 8 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (𝐴 ∈ dom (𝑟 ∖ I ) ↔ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I )))
126125notbid 319 . . . . . . 7 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (¬ 𝐴 ∈ dom (𝑟 ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I )))
127122, 1263anbi13d 1429 . . . . . 6 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )) ↔ ((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))))
128 coeq2 5722 . . . . . . . 8 (𝑠 = 𝑃 → (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃))
129128eqeq2d 2829 . . . . . . 7 (𝑠 = 𝑃 → ((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠) ↔ (𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃)))
130 difeq1 4089 . . . . . . . . 9 (𝑠 = 𝑃 → (𝑠 ∖ I ) = (𝑃 ∖ I ))
131130dmeqd 5767 . . . . . . . 8 (𝑠 = 𝑃 → dom (𝑠 ∖ I ) = dom (𝑃 ∖ I ))
132131eleq2d 2895 . . . . . . 7 (𝑠 = 𝑃 → (𝐴 ∈ dom (𝑠 ∖ I ) ↔ 𝐴 ∈ dom (𝑃 ∖ I )))
133129, 1323anbi12d 1428 . . . . . 6 (𝑠 = 𝑃 → (((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I )) ↔ ((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃) ∧ 𝐴 ∈ dom (𝑃 ∖ I ) ∧ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))))
134127, 133rspc2ev 3632 . . . . 5 ((((𝑃𝑄) ∘ 𝑃) ∈ 𝑇𝑃𝑇 ∧ ((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃) ∧ 𝐴 ∈ dom (𝑃 ∖ I ) ∧ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))) → ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))
13522, 23, 40, 42, 120, 134syl113anc 1374 . . . 4 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))
136135olcd 870 . . 3 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
13711, 136pm2.61dane 3101 . 2 ((𝜑𝐴 ∈ dom (𝑄 ∖ I )) → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
1381adantr 481 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → 𝑄𝑇)
139 coass 6111 . . . . . . 7 ((𝑄𝑃) ∘ 𝑄) = (𝑄 ∘ (𝑃𝑄))
1402, 3pmtrfcnv 18521 . . . . . . . . . 10 (𝑄𝑇𝑄 = 𝑄)
1411, 140syl 17 . . . . . . . . 9 (𝜑𝑄 = 𝑄)
142141eqcomd 2824 . . . . . . . 8 (𝜑𝑄 = 𝑄)
143142coeq2d 5726 . . . . . . 7 (𝜑 → ((𝑄𝑃) ∘ 𝑄) = ((𝑄𝑃) ∘ 𝑄))
144139, 143syl5eqr 2867 . . . . . 6 (𝜑 → (𝑄 ∘ (𝑃𝑄)) = ((𝑄𝑃) ∘ 𝑄))
1452, 3pmtrfconj 18523 . . . . . . 7 ((𝑃𝑇𝑄:𝐷1-1-onto𝐷) → ((𝑄𝑃) ∘ 𝑄) ∈ 𝑇)
14612, 31, 145syl2anc 584 . . . . . 6 (𝜑 → ((𝑄𝑃) ∘ 𝑄) ∈ 𝑇)
147144, 146eqeltrd 2910 . . . . 5 (𝜑 → (𝑄 ∘ (𝑃𝑄)) ∈ 𝑇)
148147adantr 481 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → (𝑄 ∘ (𝑃𝑄)) ∈ 𝑇)
1495coeq1d 5725 . . . . . . 7 (𝜑 → ((𝑄𝑄) ∘ (𝑃𝑄)) = (( I ↾ 𝐷) ∘ (𝑃𝑄)))
150 fcoi2 6546 . . . . . . . 8 ((𝑃𝑄):𝐷𝐷 → (( I ↾ 𝐷) ∘ (𝑃𝑄)) = (𝑃𝑄))
15135, 150syl 17 . . . . . . 7 (𝜑 → (( I ↾ 𝐷) ∘ (𝑃𝑄)) = (𝑃𝑄))
152149, 151eqtr2d 2854 . . . . . 6 (𝜑 → (𝑃𝑄) = ((𝑄𝑄) ∘ (𝑃𝑄)))
153 coass 6111 . . . . . 6 ((𝑄𝑄) ∘ (𝑃𝑄)) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄)))
154152, 153syl6eq 2869 . . . . 5 (𝜑 → (𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))))
155154adantr 481 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → (𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))))
156 f1ofn 6609 . . . . . . . . . 10 (𝑄:𝐷1-1-onto𝐷𝑄 Fn 𝐷)
15731, 156syl 17 . . . . . . . . 9 (𝜑𝑄 Fn 𝐷)
158 fnelnfp 6931 . . . . . . . . 9 ((𝑄 Fn 𝐷𝐴𝐷) → (𝐴 ∈ dom (𝑄 ∖ I ) ↔ (𝑄𝐴) ≠ 𝐴))
159157, 78, 158syl2anc 584 . . . . . . . 8 (𝜑 → (𝐴 ∈ dom (𝑄 ∖ I ) ↔ (𝑄𝐴) ≠ 𝐴))
160159necon2bbid 3056 . . . . . . 7 (𝜑 → ((𝑄𝐴) = 𝐴 ↔ ¬ 𝐴 ∈ dom (𝑄 ∖ I )))
161160biimpar 478 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → (𝑄𝐴) = 𝐴)
162 fnfvima 6986 . . . . . . . 8 ((𝑄 Fn 𝐷 ∧ dom (𝑃 ∖ I ) ⊆ 𝐷𝐴 ∈ dom (𝑃 ∖ I )) → (𝑄𝐴) ∈ (𝑄 “ dom (𝑃 ∖ I )))
163157, 77, 41, 162syl3anc 1363 . . . . . . 7 (𝜑 → (𝑄𝐴) ∈ (𝑄 “ dom (𝑃 ∖ I )))
164163adantr 481 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → (𝑄𝐴) ∈ (𝑄 “ dom (𝑃 ∖ I )))
165161, 164eqeltrrd 2911 . . . . 5 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → 𝐴 ∈ (𝑄 “ dom (𝑃 ∖ I )))
166144difeq1d 4095 . . . . . . . 8 (𝜑 → ((𝑄 ∘ (𝑃𝑄)) ∖ I ) = (((𝑄𝑃) ∘ 𝑄) ∖ I ))
167166dmeqd 5767 . . . . . . 7 (𝜑 → dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) = dom (((𝑄𝑃) ∘ 𝑄) ∖ I ))
168 f1omvdconj 18503 . . . . . . . 8 ((𝑃:𝐷𝐷𝑄:𝐷1-1-onto𝐷) → dom (((𝑄𝑃) ∘ 𝑄) ∖ I ) = (𝑄 “ dom (𝑃 ∖ I )))
16929, 31, 168syl2anc 584 . . . . . . 7 (𝜑 → dom (((𝑄𝑃) ∘ 𝑄) ∖ I ) = (𝑄 “ dom (𝑃 ∖ I )))
170167, 169eqtrd 2853 . . . . . 6 (𝜑 → dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) = (𝑄 “ dom (𝑃 ∖ I )))
171170adantr 481 . . . . 5 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) = (𝑄 “ dom (𝑃 ∖ I )))
172165, 171eleqtrrd 2913 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → 𝐴 ∈ dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ))
173 simpr 485 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → ¬ 𝐴 ∈ dom (𝑄 ∖ I ))
174 coeq1 5721 . . . . . . 7 (𝑟 = 𝑄 → (𝑟𝑠) = (𝑄𝑠))
175174eqeq2d 2829 . . . . . 6 (𝑟 = 𝑄 → ((𝑃𝑄) = (𝑟𝑠) ↔ (𝑃𝑄) = (𝑄𝑠)))
176 difeq1 4089 . . . . . . . . 9 (𝑟 = 𝑄 → (𝑟 ∖ I ) = (𝑄 ∖ I ))
177176dmeqd 5767 . . . . . . . 8 (𝑟 = 𝑄 → dom (𝑟 ∖ I ) = dom (𝑄 ∖ I ))
178177eleq2d 2895 . . . . . . 7 (𝑟 = 𝑄 → (𝐴 ∈ dom (𝑟 ∖ I ) ↔ 𝐴 ∈ dom (𝑄 ∖ I )))
179178notbid 319 . . . . . 6 (𝑟 = 𝑄 → (¬ 𝐴 ∈ dom (𝑟 ∖ I ) ↔ ¬ 𝐴 ∈ dom (𝑄 ∖ I )))
180175, 1793anbi13d 1429 . . . . 5 (𝑟 = 𝑄 → (((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )) ↔ ((𝑃𝑄) = (𝑄𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I ))))
181 coeq2 5722 . . . . . . 7 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → (𝑄𝑠) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))))
182181eqeq2d 2829 . . . . . 6 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → ((𝑃𝑄) = (𝑄𝑠) ↔ (𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄)))))
183 difeq1 4089 . . . . . . . 8 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → (𝑠 ∖ I ) = ((𝑄 ∘ (𝑃𝑄)) ∖ I ))
184183dmeqd 5767 . . . . . . 7 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → dom (𝑠 ∖ I ) = dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ))
185184eleq2d 2895 . . . . . 6 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → (𝐴 ∈ dom (𝑠 ∖ I ) ↔ 𝐴 ∈ dom ((𝑄 ∘ (𝑃𝑄)) ∖ I )))
186182, 1853anbi12d 1428 . . . . 5 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → (((𝑃𝑄) = (𝑄𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) ↔ ((𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))) ∧ 𝐴 ∈ dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I ))))
187180, 186rspc2ev 3632 . . . 4 ((𝑄𝑇 ∧ (𝑄 ∘ (𝑃𝑄)) ∈ 𝑇 ∧ ((𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))) ∧ 𝐴 ∈ dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I ))) → ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))
188138, 148, 155, 172, 173, 187syl113anc 1374 . . 3 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))
189188olcd 870 . 2 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
190137, 189pm2.61dan 809 1 (𝜑 → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wrex 3136  Vcvv 3492  cdif 3930  wss 3933  ifcif 4463  {csn 4557  {cpr 4559   cuni 4830   class class class wbr 5057   I cid 5452  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  ccom 5552   Fn wfn 6343  wf 6344  1-1-ontowf1o 6347  cfv 6348  ωcom 7569  2oc2o 8085  cen 8494  Fincfn 8497  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-fin 8501  df-pmtr 18499
This theorem is referenced by:  psgnunilem2  18552
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