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Theorem gsumval3lem1 19767
Description: Lemma 1 for gsumval3 19769. (Contributed by AV, 31-May-2019.)
Hypotheses
Ref Expression
gsumval3.b 𝐡 = (Baseβ€˜πΊ)
gsumval3.0 0 = (0gβ€˜πΊ)
gsumval3.p + = (+gβ€˜πΊ)
gsumval3.z 𝑍 = (Cntzβ€˜πΊ)
gsumval3.g (πœ‘ β†’ 𝐺 ∈ Mnd)
gsumval3.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
gsumval3.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
gsumval3.c (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
gsumval3.m (πœ‘ β†’ 𝑀 ∈ β„•)
gsumval3.h (πœ‘ β†’ 𝐻:(1...𝑀)–1-1→𝐴)
gsumval3.n (πœ‘ β†’ (𝐹 supp 0 ) βŠ† ran 𝐻)
gsumval3.w π‘Š = ((𝐹 ∘ 𝐻) supp 0 )
Assertion
Ref Expression
gsumval3lem1 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))
Distinct variable groups:   + ,𝑓   𝐴,𝑓   πœ‘,𝑓   𝑓,𝐺   𝑓,𝑀   𝐡,𝑓   𝑓,𝐹   𝑓,𝐻   𝑓,π‘Š
Allowed substitution hints:   𝑉(𝑓)   0 (𝑓)   𝑍(𝑓)
Proof of Theorem gsumval3lem1
StepHypRef Expression
1 gsumval3.h . . . . . . 7 (πœ‘ β†’ 𝐻:(1...𝑀)–1-1→𝐴)
21ad2antrr 724 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝐻:(1...𝑀)–1-1→𝐴)
3 gsumval3.w . . . . . . . . 9 π‘Š = ((𝐹 ∘ 𝐻) supp 0 )
4 suppssdm 8158 . . . . . . . . 9 ((𝐹 ∘ 𝐻) supp 0 ) βŠ† dom (𝐹 ∘ 𝐻)
53, 4eqsstri 4015 . . . . . . . 8 π‘Š βŠ† dom (𝐹 ∘ 𝐻)
6 gsumval3.f . . . . . . . . 9 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
7 f1f 6784 . . . . . . . . . 10 (𝐻:(1...𝑀)–1-1→𝐴 β†’ 𝐻:(1...𝑀)⟢𝐴)
81, 7syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐻:(1...𝑀)⟢𝐴)
9 fco 6738 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐻:(1...𝑀)⟢𝐴) β†’ (𝐹 ∘ 𝐻):(1...𝑀)⟢𝐡)
106, 8, 9syl2anc 584 . . . . . . . 8 (πœ‘ β†’ (𝐹 ∘ 𝐻):(1...𝑀)⟢𝐡)
115, 10fssdm 6734 . . . . . . 7 (πœ‘ β†’ π‘Š βŠ† (1...𝑀))
1211ad2antrr 724 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ π‘Š βŠ† (1...𝑀))
13 f1ores 6844 . . . . . 6 ((𝐻:(1...𝑀)–1-1→𝐴 ∧ π‘Š βŠ† (1...𝑀)) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š))
142, 12, 13syl2anc 584 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š))
153imaeq2i 6055 . . . . . . 7 (𝐻 β€œ π‘Š) = (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 ))
16 gsumval3.a . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ 𝑉)
176, 16fexd 7225 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 ∈ V)
18 ovex 7438 . . . . . . . . . . . 12 (1...𝑀) ∈ V
19 fex 7224 . . . . . . . . . . . 12 ((𝐻:(1...𝑀)⟢𝐴 ∧ (1...𝑀) ∈ V) β†’ 𝐻 ∈ V)
207, 18, 19sylancl 586 . . . . . . . . . . 11 (𝐻:(1...𝑀)–1-1→𝐴 β†’ 𝐻 ∈ V)
211, 20syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐻 ∈ V)
22 f1fun 6786 . . . . . . . . . . . 12 (𝐻:(1...𝑀)–1-1→𝐴 β†’ Fun 𝐻)
231, 22syl 17 . . . . . . . . . . 11 (πœ‘ β†’ Fun 𝐻)
24 gsumval3.n . . . . . . . . . . 11 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† ran 𝐻)
2523, 24jca 512 . . . . . . . . . 10 (πœ‘ β†’ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻))
2617, 21, 25jca31 515 . . . . . . . . 9 (πœ‘ β†’ ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)))
2726ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)))
28 imacosupp 8190 . . . . . . . . 9 ((𝐹 ∈ V ∧ 𝐻 ∈ V) β†’ ((Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 )))
2928imp 407 . . . . . . . 8 (((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
3027, 29syl 17 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
3115, 30eqtrid 2784 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ π‘Š) = (𝐹 supp 0 ))
3231f1oeq3d 6827 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š) ↔ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 )))
3314, 32mpbid 231 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 ))
34 isof1o 7316 . . . . 5 (𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š) β†’ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
3534ad2antll 727 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
36 f1oco 6853 . . . 4 (((𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 ) ∧ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š) β†’ ((𝐻 β†Ύ π‘Š) ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ))
3733, 35, 36syl2anc 584 . . 3 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 β†Ύ π‘Š) ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ))
38 f1of 6830 . . . . 5 (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š β†’ 𝑓:(1...(β™―β€˜π‘Š))βŸΆπ‘Š)
39 frn 6721 . . . . 5 (𝑓:(1...(β™―β€˜π‘Š))βŸΆπ‘Š β†’ ran 𝑓 βŠ† π‘Š)
4035, 38, 393syl 18 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ran 𝑓 βŠ† π‘Š)
41 cores 6245 . . . 4 (ran 𝑓 βŠ† π‘Š β†’ ((𝐻 β†Ύ π‘Š) ∘ 𝑓) = (𝐻 ∘ 𝑓))
42 f1oeq1 6818 . . . 4 (((𝐻 β†Ύ π‘Š) ∘ 𝑓) = (𝐻 ∘ 𝑓) β†’ (((𝐻 β†Ύ π‘Š) ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 )))
4340, 41, 423syl 18 . . 3 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (((𝐻 β†Ύ π‘Š) ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 )))
4437, 43mpbid 231 . 2 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ))
45 fzfi 13933 . . . . . . 7 (1...𝑀) ∈ Fin
46 ssfi 9169 . . . . . . 7 (((1...𝑀) ∈ Fin ∧ π‘Š βŠ† (1...𝑀)) β†’ π‘Š ∈ Fin)
4745, 11, 46sylancr 587 . . . . . 6 (πœ‘ β†’ π‘Š ∈ Fin)
4847ad2antrr 724 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ π‘Š ∈ Fin)
493a1i 11 . . . . . . . . 9 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ π‘Š = ((𝐹 ∘ 𝐻) supp 0 ))
5049imaeq2d 6057 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ π‘Š) = (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )))
5145a1i 11 . . . . . . . . . . . 12 (πœ‘ β†’ (1...𝑀) ∈ Fin)
528, 51fexd 7225 . . . . . . . . . . 11 (πœ‘ β†’ 𝐻 ∈ V)
5317, 52, 25jca31 515 . . . . . . . . . 10 (πœ‘ β†’ ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)))
5453ad2antrr 724 . . . . . . . . 9 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)))
5554, 29syl 17 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
5650, 55eqtrd 2772 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ π‘Š) = (𝐹 supp 0 ))
5756f1oeq3d 6827 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š) ↔ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 )))
5814, 57mpbid 231 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 ))
5948, 58hasheqf1od 14309 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (β™―β€˜π‘Š) = (β™―β€˜(𝐹 supp 0 )))
6059oveq2d 7421 . . 3 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (1...(β™―β€˜π‘Š)) = (1...(β™―β€˜(𝐹 supp 0 ))))
6160f1oeq2d 6826 . 2 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 )))
6244, 61mpbid 231 1 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  Vcvv 3474   βŠ† wss 3947  βˆ…c0 4321  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678   ∘ ccom 5679  Fun wfun 6534  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540   Isom wiso 6541  (class class class)co 7405   supp csupp 8142  Fincfn 8935  1c1 11107   < clt 11244  β„•cn 12208  ...cfz 13480  β™―chash 14286  Basecbs 17140  +gcplusg 17193  0gc0g 17381  Mndcmnd 18621  Cntzccntz 19173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287
This theorem is referenced by:  gsumval3lem2  19768
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