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| Mirrors > Home > MPE Home > Th. List > pi1addf | Structured version Visualization version GIF version | ||
| Description: The group operation of π1 is a binary operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.) |
| Ref | Expression |
|---|---|
| elpi1.g | ⊢ 𝐺 = (𝐽 π1 𝑌) |
| elpi1.b | ⊢ 𝐵 = (Base‘𝐺) |
| elpi1.1 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
| elpi1.2 | ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
| pi1addf.p | ⊢ + = (+g‘𝐺) |
| Ref | Expression |
|---|---|
| pi1addf | ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqidd 2738 | . . . . . 6 ⊢ (𝜑 → ((𝐽 Ω1 𝑌) /s ( ≃ph‘𝐽)) = ((𝐽 Ω1 𝑌) /s ( ≃ph‘𝐽))) | |
| 2 | eqidd 2738 | . . . . . 6 ⊢ (𝜑 → (Base‘(𝐽 Ω1 𝑌)) = (Base‘(𝐽 Ω1 𝑌))) | |
| 3 | fvexd 6921 | . . . . . 6 ⊢ (𝜑 → ( ≃ph‘𝐽) ∈ V) | |
| 4 | ovexd 7466 | . . . . . 6 ⊢ (𝜑 → (𝐽 Ω1 𝑌) ∈ V) | |
| 5 | elpi1.g | . . . . . . . 8 ⊢ 𝐺 = (𝐽 π1 𝑌) | |
| 6 | elpi1.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
| 7 | elpi1.2 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
| 8 | eqid 2737 | . . . . . . . 8 ⊢ (𝐽 Ω1 𝑌) = (𝐽 Ω1 𝑌) | |
| 9 | elpi1.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺) | |
| 10 | 9 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
| 11 | 5, 6, 7, 8, 10, 2 | pi1blem 25072 | . . . . . . 7 ⊢ (𝜑 → ((( ≃ph‘𝐽) “ (Base‘(𝐽 Ω1 𝑌))) ⊆ (Base‘(𝐽 Ω1 𝑌)) ∧ (Base‘(𝐽 Ω1 𝑌)) ⊆ (II Cn 𝐽))) |
| 12 | 11 | simpld 494 | . . . . . 6 ⊢ (𝜑 → (( ≃ph‘𝐽) “ (Base‘(𝐽 Ω1 𝑌))) ⊆ (Base‘(𝐽 Ω1 𝑌))) |
| 13 | 1, 2, 3, 4, 12 | qusin 17589 | . . . . 5 ⊢ (𝜑 → ((𝐽 Ω1 𝑌) /s ( ≃ph‘𝐽)) = ((𝐽 Ω1 𝑌) /s (( ≃ph‘𝐽) ∩ ((Base‘(𝐽 Ω1 𝑌)) × (Base‘(𝐽 Ω1 𝑌)))))) |
| 14 | 5, 6, 7, 8 | pi1val 25070 | . . . . 5 ⊢ (𝜑 → 𝐺 = ((𝐽 Ω1 𝑌) /s ( ≃ph‘𝐽))) |
| 15 | 5, 6, 7, 8, 10, 2 | pi1buni 25073 | . . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 = (Base‘(𝐽 Ω1 𝑌))) |
| 16 | 15 | sqxpeqd 5717 | . . . . . . 7 ⊢ (𝜑 → (∪ 𝐵 × ∪ 𝐵) = ((Base‘(𝐽 Ω1 𝑌)) × (Base‘(𝐽 Ω1 𝑌)))) |
| 17 | 16 | ineq2d 4220 | . . . . . 6 ⊢ (𝜑 → (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) = (( ≃ph‘𝐽) ∩ ((Base‘(𝐽 Ω1 𝑌)) × (Base‘(𝐽 Ω1 𝑌))))) |
| 18 | 17 | oveq2d 7447 | . . . . 5 ⊢ (𝜑 → ((𝐽 Ω1 𝑌) /s (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))) = ((𝐽 Ω1 𝑌) /s (( ≃ph‘𝐽) ∩ ((Base‘(𝐽 Ω1 𝑌)) × (Base‘(𝐽 Ω1 𝑌)))))) |
| 19 | 13, 14, 18 | 3eqtr4d 2787 | . . . 4 ⊢ (𝜑 → 𝐺 = ((𝐽 Ω1 𝑌) /s (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)))) |
| 20 | phtpcer 25027 | . . . . . 6 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽) | |
| 21 | 20 | a1i 11 | . . . . 5 ⊢ (𝜑 → ( ≃ph‘𝐽) Er (II Cn 𝐽)) |
| 22 | 11 | simprd 495 | . . . . . 6 ⊢ (𝜑 → (Base‘(𝐽 Ω1 𝑌)) ⊆ (II Cn 𝐽)) |
| 23 | 15, 22 | eqsstrd 4018 | . . . . 5 ⊢ (𝜑 → ∪ 𝐵 ⊆ (II Cn 𝐽)) |
| 24 | 21, 23 | erinxp 8831 | . . . 4 ⊢ (𝜑 → (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) Er ∪ 𝐵) |
| 25 | eqid 2737 | . . . . 5 ⊢ (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)) | |
| 26 | eqid 2737 | . . . . 5 ⊢ (+g‘(𝐽 Ω1 𝑌)) = (+g‘(𝐽 Ω1 𝑌)) | |
| 27 | 5, 6, 7, 10, 25, 8, 26 | pi1cpbl 25077 | . . . 4 ⊢ (𝜑 → ((𝑎(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑐 ∧ 𝑏(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))𝑑) → (𝑎(+g‘(𝐽 Ω1 𝑌))𝑏)(( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))(𝑐(+g‘(𝐽 Ω1 𝑌))𝑑))) |
| 28 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ ∪ 𝐵 ∧ 𝑑 ∈ ∪ 𝐵)) → 𝐽 ∈ (TopOn‘𝑋)) |
| 29 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ ∪ 𝐵 ∧ 𝑑 ∈ ∪ 𝐵)) → 𝑌 ∈ 𝑋) |
| 30 | 8, 28, 29 | om1plusg 25067 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ ∪ 𝐵 ∧ 𝑑 ∈ ∪ 𝐵)) → (*𝑝‘𝐽) = (+g‘(𝐽 Ω1 𝑌))) |
| 31 | 30 | oveqd 7448 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ∪ 𝐵 ∧ 𝑑 ∈ ∪ 𝐵)) → (𝑐(*𝑝‘𝐽)𝑑) = (𝑐(+g‘(𝐽 Ω1 𝑌))𝑑)) |
| 32 | 15 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ ∪ 𝐵 ∧ 𝑑 ∈ ∪ 𝐵)) → ∪ 𝐵 = (Base‘(𝐽 Ω1 𝑌))) |
| 33 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ ∪ 𝐵 ∧ 𝑑 ∈ ∪ 𝐵)) → 𝑐 ∈ ∪ 𝐵) | |
| 34 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ ∪ 𝐵 ∧ 𝑑 ∈ ∪ 𝐵)) → 𝑑 ∈ ∪ 𝐵) | |
| 35 | 8, 28, 29, 32, 33, 34 | om1addcl 25066 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ ∪ 𝐵 ∧ 𝑑 ∈ ∪ 𝐵)) → (𝑐(*𝑝‘𝐽)𝑑) ∈ ∪ 𝐵) |
| 36 | 31, 35 | eqeltrrd 2842 | . . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ ∪ 𝐵 ∧ 𝑑 ∈ ∪ 𝐵)) → (𝑐(+g‘(𝐽 Ω1 𝑌))𝑑) ∈ ∪ 𝐵) |
| 37 | pi1addf.p | . . . 4 ⊢ + = (+g‘𝐺) | |
| 38 | 19, 15, 24, 4, 27, 36, 26, 37 | qusaddf 17599 | . . 3 ⊢ (𝜑 → + :((∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))) × (∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))))⟶(∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)))) |
| 39 | 5, 6, 7, 10, 25 | pi1bas3 25076 | . . . . 5 ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)))) |
| 40 | 39 | sqxpeqd 5717 | . . . 4 ⊢ (𝜑 → (𝐵 × 𝐵) = ((∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))) × (∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))))) |
| 41 | 40 | feq2d 6722 | . . 3 ⊢ (𝜑 → ( + :(𝐵 × 𝐵)⟶(∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))) ↔ + :((∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))) × (∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))))⟶(∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))))) |
| 42 | 38, 41 | mpbird 257 | . 2 ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶(∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵)))) |
| 43 | 39 | feq3d 6723 | . 2 ⊢ (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐵 ↔ + :(𝐵 × 𝐵)⟶(∪ 𝐵 / (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))))) |
| 44 | 42, 43 | mpbird 257 | 1 ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∪ cuni 4907 × cxp 5683 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Er wer 8742 / cqs 8744 Basecbs 17247 +gcplusg 17297 /s cqus 17550 TopOnctopon 22916 Cn ccn 23232 IIcii 24901 ≃phcphtpc 25001 *𝑝cpco 25033 Ω1 comi 25034 π1 cpi1 25036 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-qus 17554 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-cn 23235 df-cnp 23236 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-ii 24903 df-htpy 25002 df-phtpy 25003 df-phtpc 25024 df-pco 25038 df-om1 25039 df-pi1 25041 |
| This theorem is referenced by: (None) |
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