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Mirrors > Home > MPE Home > Th. List > 2sqreulem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for 2sqreu 26032 et. (Contributed by AV, 25-Jun-2023.) |
Ref | Expression |
---|---|
2sqreulem4.1 | ⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
Ref | Expression |
---|---|
2sqreulem4 | ⊢ ∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqreulem3 26029 | . . . 4 ⊢ ((𝑎 ∈ ℕ0 ∧ (𝑏 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0)) → (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) | |
2 | 1 | ralrimivva 3191 | . . 3 ⊢ (𝑎 ∈ ℕ0 → ∀𝑏 ∈ ℕ0 ∀𝑐 ∈ ℕ0 (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) |
3 | 2sqreulem4.1 | . . . . 5 ⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | |
4 | 3 | rmobii 3396 | . . . 4 ⊢ (∃*𝑏 ∈ ℕ0 𝜑 ↔ ∃*𝑏 ∈ ℕ0 (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
5 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑏ℕ0 | |
6 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑐ℕ0 | |
7 | nfsbc1v 3792 | . . . . . 6 ⊢ Ⅎ𝑏[𝑐 / 𝑏]𝜓 | |
8 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑏((𝑎↑2) + (𝑐↑2)) = 𝑃 | |
9 | 7, 8 | nfan 1900 | . . . . 5 ⊢ Ⅎ𝑏([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃) |
10 | sbceq1a 3783 | . . . . . 6 ⊢ (𝑏 = 𝑐 → (𝜓 ↔ [𝑐 / 𝑏]𝜓)) | |
11 | oveq1 7163 | . . . . . . . 8 ⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2)) | |
12 | 11 | oveq2d 7172 | . . . . . . 7 ⊢ (𝑏 = 𝑐 → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + (𝑐↑2))) |
13 | 12 | eqeq1d 2823 | . . . . . 6 ⊢ (𝑏 = 𝑐 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) |
14 | 10, 13 | anbi12d 632 | . . . . 5 ⊢ (𝑏 = 𝑐 → ((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃))) |
15 | 5, 6, 9, 14 | rmo4f 3726 | . . . 4 ⊢ (∃*𝑏 ∈ ℕ0 (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∀𝑏 ∈ ℕ0 ∀𝑐 ∈ ℕ0 (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) |
16 | 4, 15 | bitri 277 | . . 3 ⊢ (∃*𝑏 ∈ ℕ0 𝜑 ↔ ∀𝑏 ∈ ℕ0 ∀𝑐 ∈ ℕ0 (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) |
17 | 2, 16 | sylibr 236 | . 2 ⊢ (𝑎 ∈ ℕ0 → ∃*𝑏 ∈ ℕ0 𝜑) |
18 | 17 | rgen 3148 | 1 ⊢ ∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃*wrmo 3141 [wsbc 3772 (class class class)co 7156 + caddc 10540 2c2 11693 ℕ0cn0 11898 ↑cexp 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 |
This theorem is referenced by: 2sqreunnlem2 26031 2sqreu 26032 2sqreult 26034 2sqreultb 26035 |
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